## Graph Of Sine And Cosine Functions Real World Applications

The cosine is known as an even function, and the sine is known as an odd function. Identities and equations do not include any half-angle relationships. Objectives: To graph the sine and cosine functions; to identify the graphs of the sine and cosine functions. The amplitude is 2. Are you looking to apply what you learn in the classroom to the real world? Trigonometry is a subject that has lots of practical applications. Markov processes are widely used in performance modelling of wireless and mobile communication systems. Frequency is the number of cycles in a given unit of time, so it is the reciprocal of the period of a function. Whenever you see an "oscilloscope," for example when you play music using certain programs on a computer, you're really seeing a whole bunch of sine waves added together. In that program you used the built-in sine and cosine functions. Frequency is the number of cycles in a given unit of time. Fortunately, we can analyze the problem by first representing it as a linear function and then interpreting the components of the function. 762-768) optional 2 optional 1 • Graph trigonometric functions. So the graph looks like a very simple wave. One method to write a sine or cosine function that models a sinusoid is to fi nd the values of a, b, h, and k for y = a sin b(x − h) + k or y = a cos b(x − h) + k where ∣ a ∣ is the amplitude, — 2π is the period. The inverses of sine, cosine and tangent functions are not functions unless the domains are limited. Request PDF | A Discrete Sine Cosine Algorithm for Community Detection | As an important research area in complex networks, community detection problems (CDPs) have aroused widespread concern. Find an expression for a volume. Evaluate Inverse Trigonometric Functions v. We often graph sine over time (so we don't write over ourselves) and sometimes the "thing" doing sine is also moving, but this is optional! A spring in one dimension is a perfectly happy sine wave. acute triangles using the sine law and the cosine law, including problems arising from real-world applications; demonstrate an understanding of periodic relationships and the sine function, and make connections between the numeric, graphical, and algebraic representations of sine functions; identify and represent sine functions, and. Using your guidelines, you should be able to sketch in a smooth curve: see figure 3. FIND the coordinates of A & C. The graphs of sine, cosine, and tangent. ) like any other function with a domain or real numbers!. • Solve real world and mathematical problems using a system of two first-degree equations in two variables. First, take a look the Taylor series representation of exponential function, and trigonometric functions, sine, and cosine,. The vertical shift is 2. Find an equation for a sine function that has amplitude of 4 a period of π. 9) Graph sine and cosine functions (PC-N. THE COSINE FUNCTION You can graph the cosine function y = cos x. Trig Functions: The Ferris Wheel Applications of Trig Functions - Duration: 13:37. txt) or view presentation slides online. 25 scaffolded questions on equation graph involving amplitude and periodplus model problems explained step by step. Find Period and Amplitude 2. Whenever you see an "oscilloscope," for example when you play music using certain programs on a computer, you're really seeing a whole bunch of sine waves added together. Determine the values for angles between 0° and 360°. Trigonometry An In Depth Approach to Sine, Cosine, Tangent, Cotangent, Secant and Cosecant: Trigonometry Ratios and Their Graphs and Real World Applications. Recall that the sine and cosine functions relate real number values to the x– and y-coordinates of a point on the unit circle. Lesson 9-11: Graphing Trigonometric Functions and Applications in Real World Contexts Learning Goals #8, 9, 10: How do I use the critical values of the Sine and Cosine curve to assist in in graphing Sine and Cosine functions? How do I contextualize Trig Functions in real life situations? Warm-Up! WITH A PARTNER. I can identify the important features (amplitude, period, max, min, x-intercepts, and frequency) of the sine, cosine, and tangent functions. Sinusoidal functions graph wave forms. Graphs of the sine and cosine functions; and Periodic Behavior. The reality is that the functions of sine, cosine and tangent are embedded in the foundations of modern mathematics and, as you’ll discover, the world around us. Students use special triangles to geometrically determine the values of sine. KEY STANDARDS ADDRESSED: MA3A3. We start by revisiting the Ferris wheel. Write a rule in the form f(t) = A sin Bt that expresses this oscillation where t represents the number of seconds. Students are expected to be able to interpret functions in the context of a problem, write functions, and apply functions to periodic phenomenon. The hyperbolic sine and the hyperbolic cosine are. If we recall from the previous section we said that $$f\left( x \right)$$ is nothing more than a fancy way of writing $$y$$. Includes full solutions and score reporting. Line 1 just restates Euler's formula. So for example, if x = 2, the y-value will be y = 2 sin. Accelerated Mathematics III - Unit 5 Investigating Trigonometry Graphs Student Edition INTRODUCTION: In this unit students develop an understanding of the graphs of the six trigonometric functions: sine, cosine, tangent, cotangent, cosecant, and secant. Graphing Sine & Cosine w/out a Calculator Pt 2. Read: 'The sine of thirty degrees is zero point five. Trigonometry means the study of the triangle. Verify the size of a specified angle and hence that two triangles within the given diagram are similar. The resulting graph will be displayed as you change the values. ( Source: Wikipedia, try not to get hypnotized. Find angle pi on a grid. 387 In-class/Homework: pg. Apply right-angle trigonometry to a scenario. Many compression algorithms like JPEG use fourier transforms that rely on sin and cos. Sine and Cosine Part 2 - given the graph of a sinusoidal function, determine an equation that defines it D2. We offer targeted baccalaureate programs of study, pre-baccalaureate programs of study for transfer, associate of arts and associate of science degrees, and serve as a portal to graduate education. By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval$\,\left[-1,1\right]. - He excelled in the trigonometric functions unit where he was able to sketch sinusoidal functions using the transformations of key points of the sine or. Different sounds create different waves. Use sinusoidal functions to solve problems. In this example, we are multiplying the sine of each x-value by the x-value. Transformations of Trigonometric Functions. " For example, "an oscilloscope is an electronic instrument used to display changing electrical signals. and “use sine and cosine functions to model real-life data,” iii. To be able to solve real-world problems using the Law of Sines and the Law of Cosines This tutorial reviews two real-world problems, one using the Law of Sines and one using the Law of Cosines. We do: Writing Sinusoidal Functions from a Graph. march 21, 2007. To find the cosine of angle pi, you need graph paper. Moreoever, the pattern repeats, so this is still a periodic function. can be used to assign a particular use of the plot function to a particular figure wi. Assessments Summative: Tests and benchmark. ppt), PDF File (. Markov processes are widely used in performance modelling of wireless and mobile communication systems. Graphing a Sinusoidal Function. ( Source: Wikipedia, try not to get hypnotized. A periodic function is defined as a function that repeats its values in regular periods. Trig Functions: The Ferris Wheel Arnold Tutoring Solving Problems with the Sine and Cosine Functions - Lesson - Duration: Determining the Equation of a Sine and Cosine Graph - Duration:. For this, the phase shift will be 172. So what do they look like on a graph on a coordinate plane? Let’s start with the sine function. Sketch translations of graphs of sine and cosine functions; use sine and cosine functions to model real-life data. 2 Graph lines (including those of the form x = h and y = k) given appropriate information. I am confused as I am unsure whether time (e. As we can see in Figure 6, the sine function is symmetric about the origin. In fact, there is a zero derivative at this point here, which leads us to believe that maybe what we should have done was to have broken this curve down into the union of two one-to-one functions. y = 2 sin 2x 14. Amplitude and Period for Sine and Cosine Functions Worksheet Author: marthasweeney Last modified by: Leake, Sherry LHS-STAFF Created Date: 6/13/2016 5:56:00 PM. 2]Solve right and oblique triangles including application problems. Trigonometric formulae are useful for solving problems in two dimensions. Find all values of A and B such that. PERIODIC FUNCTIONS A. Four Function Scientific. Lesson 12: Ferris Wheels—Using Trigonometric Functions to Model Cyclical Behavior Student Outcomes Students review how changing the parameters 𝐴, 𝜔, ℎ, and 𝑘 in ( )=𝐴sin(𝜔( −ℎ))+𝑘 affects the graph of a sinusoidal function. 28rad has the same cosine as 5rad, they're inverse angles. php(143) : runtime-created function(1) : eval()'d code(156) : runtime-created. Solve real world applications problems using systems of equations and inequalities Graph rational functions with asymptotes Graph sine and cosine curves. How the values of A and B affect the shape of the graph y = A sin(Bx). The table of values for the functions. The number π (pi)— or at least the closest approximation that fits in a JavaScript number—is available as Math. Graphing Sine & Cosine w/out a Calculator Pt 2. When they notice that these points are already on the graph, make a note about the cyclic nature of sine and cosine. List of trigonometric identities 2 Trigonometric functions The primary trigonometric functions are the sine and cosine of an angle. Correctness of a skewed cosine similarity graph I am currently implementing a word2vec model that uses the cosine similarity to determine the similarity between two vectors. A great discussion occurs when the students are asked to justify the function they want to use. The simplest one is y = sin(x). sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number. Topic:Real world application of sine graphs. Graph) 3 2 f x = ( ) 2sin(x. Graphing Trigonometric Functions Real-World Applications of Trigonometric Functions. In trig speak, you say something like this: If theta represents all the angles in the domain of the two functions. ) Circles are an example of two sine waves. The reality is that the functions of sine, cosine and tangent are embedded in the foundations of modern mathematics and, as you’ll discover, the world around us. In maths, you have real life applications on any thing that you study. For now, focus primarily on sine, cosine, and tangent. Trigonometric Functions, you will begin by learning about the inverses of quadratics and other functions. any anual average can be repeated year to year. Graphing Sine and Cosine Functions. Les't compare with. Graph a sine or cosine function with a given phase shift, period, and amplitude. When plotting all the possible cosine similarities, I get the following. find the equation of a sinusoidal function given its graph or its properties. 7m, but I am unsure how to interval the axis. Of course, this was to be expected, but this isn't the real problem; the real problem is that if your function has some crucial points it must pass through (which is certainly true for trigonometry functions), the truncation will move the curve away from those points. So for example, if x = 2, the y-value will be y = 2 sin. Intro Tangent & Cotangent Graphs. Outcome 6: Set up and solve exponential and logarithmic equations; then identify and sketch graphs of the functions. 4 5 Graphing the Trigonometric Functions (Part 2) 2 The sine function Imagine a particle on the unit circle, starting at (1,0) and rotating counterclockwise around the origin. How to find and the slopes of lines to write and graph linear equations in two variables. Given an expression for the time rate of change of a quantity, find the initial rate. 762 Chapter 14 Trigonometric Graphs and Identities • Graph trigonometric functions. Sample Problem. 1 solve problems, including those that arise from real-world applications (e. graph the sine and cosine functions with varied amplitudes and periods. 14-1 Graphs of Sine and Cosine Example 3: Sound Application Use a sine function to graph a sound wave with a period of 0. Those functions were predefined for you, meaning that you didn't have to tell the computer how to compute the sine and cosine of an angle. Focus on creating a graph and formula from a story (starting from time 6:55) You Do Now: Warmup - Sketching a Cosine Wave. When the radian measure of angles was introduced in the early eighteenth century by Roger Cotes and the trigonometric ratios of angles in the plane began to be viewed as functions of a linearized angle, the undulating movement of the sine and cosine graphs fed the. Les't compare with. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector. Trigonometric identities like finding. Graphing Secant & Cosecant. A real life example of the sine function could be a ferris wheel. Graphing Sine and Cosine Trig Functions With Transformations, Phase Shifts, Period - Domain & Range - Duration: 18:35. dilations, and horizontal and vertical translations of functions. This example shows how to use both CORDIC-based and lookup table-based algorithms provided by the Fixed-Point Designer™ to approximate the MATLAB® sine (SIN) and cosine (COS) functions. Every position of the particle corresponds with an angle, ?, where y sin ?. In line 3 we plug in -x into Euler's formula. Example: if your cutting down a tree and need to know where it will. Using f (x) = cos x as a guide, graph the function h (x) = __1 cos 2 3 x. SC-Common Core Precalculus Scope and Sequence Unit Topic Lesson Lesson Objectives Functions Functions and Graphs Graphing Linear Equations Find the slope of a line through two points Find the x- and y-intercepts of a line Find zeros of linear functions Graph linear equations Operations with Functions Find composite functions. Graph Translations 3. Plus ITEM TYPES: MC. Determine the characteristics of the graphs of the six basic trigonometric functions. I can graph the sine, cosine, and tangent functions. 3 make connections between the sine ratio and the sine function and between the cosine ratio and the cosine function by graphing the relationship between angles from 0º to 360º and the corresponding sine ratios or cosine ratios, defining this relationship as the function f(x) =sinx or f(x) =cosx, and explaining why the relationship is a function. ( Source: Wikipedia, try not to get hypnotized. Equation of Sine and Cosine from a Graph. Calculus is made up of Trigonometry and Algebra. inverse MTH 112 Elementary Functions Chapter 5 The Trigonometric Functions -. You can use the graph for a trigonometry function to show average daily temperatures at a specific location. Most radio communication is based on the use of combinations of sines and cosine waves. To understand how the absolute value can be applied to the real world, we’ll look at two topics: What is considered normal with regard to the temperature of the human body and; Fuel economy of two vehicles: a Honda Odyssey and a Nissan Altima. Trig also has applications in fields as broad as financial analysis, music theory, biology, medical imaging, cryptology, game development, and seismology. Amplitude is first, then sine or cosine, then B, which you get by doing 2pi/10 (period), which reduces to pi/5, then it is the parenthesis with the transformation, and then last but not least is the sinusodial axis. Application: Sine and cosine functions can be used to model real-world phenomena, such as sound waves. In the chapter on Trigonometric Functions, we examined trigonometric functions such as the sine function. functions; increasing/decreasing functions; finding intercepts with axes. Evaluate trigonometric functions of real numbers. Modeling Using Sinusoidal Functions Sine and cosine functions model many real-world situations. The graph of the sine function is called the sine curve. Defining Sine and Cosine Functions. When they notice that these points are already on the graph, make a note about the cyclic nature of sine and cosine. Amplitude is first, then sine or cosine, then B, which you get by doing 2pi/10 (period), which reduces to pi/5, then it is the parenthesis with the transformation, and then last but not least is the sinusodial axis. Topic:Real world application of sine graphs. Example: if your cutting down a tree and need to know where it will. Find all values of A and B such that. 7m, but I am unsure how to interval the axis. This shape is also called a sine. Evaluate Inverse Trigonometric Functions v. For large arguments we would expect these functions to be a quarter period out of phase, just like cosine and sine, since asymptotically J 1 is proportional to cos(z – 3π/4) / √z and Y 1 is proportional to sin(z – 3π/4) / √z. Graphing Sine & Cosine w/out a Calculator Pt1. The function is a model for the hours of daylight in Center City. Sum and Difference, Double Angle and Half Angle. The size of a hyperbolic angle is twice the area of its hyperbolic sector. Students will use the model to make a prediction about a missing piece of data. 2 cos 2 cos 2 cos 2 sin( ) cos Now we can clearly see that if we horizontally shift the cosine function to the right by π/2 we get the sine function. 1 works in a wide range of settings: any time we have an exponential equation of the form $$p \cdot q^t + r = s\text{,}$$ we can solve for $$t$$ by first isolating the exponential expression $$q^t$$ and then by taking the natural logarithm of both sides of the equation. In maths, you have real life applications on any thing that you study. The cosine is known as an even function, and the sine is known as an odd function. Cumulative Review. Along the way, youll get plenty of practice, from fully guided examples to independent end-of-chapter drills and test-like samples. Midline Amplitude And Period Review Article Khan Academy Solving Problems Using Sinusoidal Functions Real World Applications Ferris Wheel Trig Example Youtube Period Of Cosine Function Math Ladyprestige Club. Next, lets take the 5 from the front. 2 Define sine and cosine as functions of the radian measure of an angle in terms of the x- and y-coordinates of the point on the unit circle. But just as you could make the basic quadratic, y = x 2, more complicated, such as y = -(x + 5) 2 - 3, so also trig graphs can be made more complicated. I know that the amplitude of the function is 1. When I consider how to address the Precalculus objectives "to solve real-life problems involving harmonic motion" ii. A cosine function will start at its minimum or maximum value on the y-axis; however, the sine graph will start at its midline on the y-axis. Graph sine and cosine functions with various amplitudes and periods. 6 Modeling with Trigonometric Functions 507 Writing Trigonometric Functions Graphs of sine and cosine functions are called sinusoids. Determine the values for angles between 0° and 360°. can be used to assign a particular use of the plot function to a particular figure wi. A sine wave is what you get when you plot the sine function on a graph. Psst… don’t over-focus on a single diagram, thinking tangent is always smaller than 1. Defining Sine and Cosine Functions. Sketch the graphs of basic sine and cosine functions; use amplitude and period to help sketch the graphs of sine and cosine functions. Phase Shift = to the right Vertical Shift = 4 4. How to analyze graphs of functions. Sine, Cosine, Tangent Applications. My kids have seen this a bunch so this should only take a minute or two. 2]Solve right and oblique triangles including application problems. Analysis: I developed this lesson plan so that students would be introduced to graphs…. Aug 15, 2015 · C2 Sine and Cosine Rule - 1 - Basic Introduction (AS Maths A Level trigonometry) ukmathsteacher. In line 4 we use the properties of cosine (cos -x = cos x) and sine (sin -x = -sin x) to simplify the. Graphing the sine and cosine functions is a topic that students will carry on with them throughout the rest of their future science and mathematics courses. Graphs of tan, cot, sec and csc; 5. Real life word problems that involve the sine and cosine function can be used to keep the students engaged in the topic. inverse MTH 112 Elementary Functions Chapter 5 The Trigonometric Functions -. The inverses of sine, cosine and tangent functions are not functions unless the domains are limited. Many compression algorithms like JPEG use fourier transforms that rely on sin and cos. Investigate Transformations of Sine and Cosine Functions Many real-world processes can be modelled with sinusoidal functions. functions for special angles using the unit circle and right triangle definitions. Evaluating Inverse. - She did an excellent job in the trigonometry unit where she was able to use the sine and cosine laws to solve problems on both her assignment and. We can prove this by using the cofunction identity and the negative angle identity for cosine. Lesson 6-6 Modeling Real-World Data with Sinusoidal Functions 389 Graphing Calculator Tip For keystroke instruction on how to find sine regression statistics, see page A25. Amplitude and Period for Sine and Cosine Functions Worksheet Author: marthasweeney Last modified by: Leake, Sherry LHS-STAFF Created Date: 6/13/2016 5:56:00 PM. The sine function “starts” at the midline, but the leading negative causes it to reflect across the midline. Course Objectives: * Students will be able to develop and understand transferable pre-calculus skills necessary for success in Precalculus and beyond * Students will be able to identify, evaluate, perform operations on, and find the domain, range, and inverse of functions * Students will be able to draw common graphs along with transformations and reflections * Students will be able to create. Use sinusoidal functions to solve problems. The heat equation is used to model how things get hot (electronics, spacecraft, ovens, etc). Find the amplitude, period, and phase shift of a sine or cosine functions from its equation. Sine and cosine functions can be used to model many real-life scenarios –radio waves, tides, musical tones, electrical currents. Frequency is the number of cycles in a given. Analytic Trigonometry. Sine and cosine waves can be applied to all sinusoid problems of the real-world. Being a cofunction, means that complementary input angles leads to the same output , as shown in the following example: Problem 2. Students will investigate and use the graphs of the six trigonometric functions. 1 RTP Scenarios and Terminology 12. 2 RTP Packet Format 12. Understanding Basic Sine & Cosine Graphs. c) sketch the graph of the function by using transformations for at least a two-period interval; and d) investigate the effect of changing the parameters in a trigonometric function on the graph of the function. When teaching students the value of sinusoidal models and graphs, such as sine and cosine functions, students often feel like they must memorize formulas and apply them with little recollection or understanding. Graph y = sin x. Using f (x) = cos x as a guide, graph the function h (x) = __1 cos 2 3 x. In particular: The slope of a graph is related to how fast the quantity being graphed is changing. Graphing Sine and Cosine Functions. The Sine, Cosine and Tangent functions are often applied to real world scenarios. C8 demonstrate an understanding of real-world relationships by translating between graphs, tables, and written descriptions. Third, this is a sine graph with a leading negative. Hence, we start this lesson by recalling these functions and their corresponding graphs. functions, and graph them, with and without the use of digital technologies (ACMMG274) prove that the. Trigonometry can be used to roof a house, to make the roof inclined ( in the case of single individual bungalows) and the height of the roof in buildings etc. Figure 4 The cosine function Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. Graphs of the Trigonometric Functions; 1. The Code of Federal Regulations is a codification of the general and permanent rules published in the Federal Register by the Executive departments and agencies of the Federal Government. Write a cosine function that describes the height of the nail above ground as a function of angular distance. Trigonometric functions (Level C) Radian measure (including what are radians, converting from degrees to radians, converting from radians to degrees, using radians measure in real world applications) Graphs of sine, cosine, and tangent functions. The sine function. A single note can be modeled on a sine curve, and a chord can be modeled with multiple sine curves used in conjunction with one another. Special triangles can be used to geometrically determine the values of the six trigonometric functions at ,, 6 4 3 , and their integral multiples. Duration: 2 Day(s) Learning Targets. Applications of Sine and Cosine Graphs Learning Task: Trigonometry functions are often used to model periodic data. Key Questions. SheLovesMath. Graphing Sine and Cosine Functions. Chapter Five – Trigonometric Functions What to Expect: A new formula designed for sin and cos graphs, details about frequencies, new vocabulary, explaining periodic trends, and the like. Notes Chapter 6 Section 6 ( Modeling Real-World Data with Sinusoidal Functions ) -pg. Section 3-5 : Graphing Functions. Trigonometric Functions By the end of this course, students will: - solve problems involving trigonometry in acute triangles using the sine law and the cosine law, including problems arising from real-world applications; - demonstrate an understanding of periodic relationships and the sine function, and make connections between the identify and. The cosine of an angle is the ratio of x r {\displaystyle {\tfrac {x}{r}}} , so in the unit circle, the cosine is the x -coordinate of the point of rotation. - She did an excellent job in the trigonometry unit where she was able to use the sine and cosine laws to solve problems on both her assignment and. C2 create and analyse scatter plots of periodic data. Press the button marked "sin" to change to the cosine. 2-use technology, graphs, and tables to compare sine graphs. 3 Finding Maximum and Minimum Values and Zeros of Sine and Cosine 2. GCSE Maths Pythagoras and Trigonometry Level 8-9. These functions are most conveniently defined in terms of the exponential function, with sinh z = 1 / 2 (e z − e −z) and cosh z = 1 / 2 (e z + e. 391-392 #'s 7-16 all I will be able to: Model real-world data using sine and cosine functions. 8 Graphs, Inverses, and Applications of Trigonometric Functions. Find the amplitude and period of the function. Applications of Exponential Functions The best thing about exponential functions is that they are so useful in real world situations. If the set of input values is a ﬁnite set then the models are known as discrete models. When teaching students the value of sinusoidal models and graphs, such as sine and cosine functions, students often feel like they must memorize formulas and apply them with little recollection or understanding. 7 The student will identify the domain and range of the inverse trigonometric functions and recognize the graphs of these functions. So, temperature as a function of days. A real life example of the sine function could be a ferris wheel. Sample records for nino-southern oscillation eventsnino-southern oscillation events «. To be able to solve real-world problems using the Law of Sines and the Law of Cosines This tutorial reviews two real-world problems, one using the Law of Sines and one using the Law of Cosines. and "use sine and cosine functions to model real-life data," iii. Find the amplitude, period, and phase shift of a sine or cosine functions from its equation. sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number. Write the sum and difference identities for sine, cosine, and tangent. Chapter 4 Sections 4. This means that bodies. On the ground, from the object you measure off a know distant in a straight line from the base. 2) Watch the video and pay attention to the key ideas listed. If you take a rope, fix the two ends, and let it hang under the force of gravity, it will naturally form a hyperbolic cosine curve. The sine and cosine of an angle are both circular functions, and they are the fundamental circular functions. Water Depth Word Problem Modeled with Cosine Sine Function. 5/175* share an equivalent sine. Namely, if we look at the graph 'y' equals hyperbolic cosine 'x', observe that whereas the function is single valued, it is not one-to-one. Find patterns in given array of numbers. The approach used in Example 3. 2 Define sine and cosine as functions of the radian measure of an angle in terms of the x- and y-coordinates of the point on the unit circle. Find the frequency in hertz for this sound wave. A single note can be modeled on a sine curve, and a chord can be modeled with multiple sine curves used in conjunction with one another. For large arguments we would expect these functions to be a quarter period out of phase, just like cosine and sine, since asymptotically J 1 is proportional to cos(z – 3π/4) / √z and Y 1 is proportional to sin(z – 3π/4) / √z. Frequency is the number of cycles in a given unit of time. Sineing on to the job Since we know that a triangle has 180 degrees, we can subtract 56 degrees and 91 degrees from it to find our missing angle Using the law of sines we can then set up this equation sin 91 degrees/ xft = sin 33/6ft After crossmultipying and then dividing to. Now we must stretch out the amplitude of the sine graph. Les't compare with. So, temperature as a function of days. This is an example of how a Graph of a Cosine Function looks like:. Animated gifs showing relationships between functions and geometry eg trigonometric functions How the unit circle gives us the sine wave Teaching math with the Drum Loop Real sinusoid on a timebase, formed by a linear increment of complex argument in time. Therefore, the period of this graph is 2π/(1/3) which is 6π. To find the period of a sine graph, take the B value, which is 2, and divide 2π by it. Description: To be able to solve real-world problems using the Law of Sines and the Law of Cosines. Land surveying makes an extensive use of the sine and cosine law. The basic sine and cosine graphs C. Seymour Public Schools Curriculum Subject or course name 5 graph the basic sine and cosine functions. Do not forget this. state the properties for the Sine and Cosine graph identify the values of a, k, c and d and apply it to a sinusoidal function to graph the resulting transformation Lesson:. Graph Trigonometric Functions (1), cosine function with solution. ' Below is a listing of several popular input and output values for the three main trigonometry functions. 1) Graph sine functions (PC-N. Graphing Sine and Cosine Trig Functions With Transformations, Phase Shifts, Period - Domain & Range - Duration: 18:35. 1: Graphs of the Sine and Cosine Functions. Water Depth Word Problem Modeled with Cosine Sine Function. One method to write a sine or cosine function that models a sinusoid is to fi nd the values of a, b, h, and k for y = a sin b(x − h) + k or y = a cos b(x − h) + k where ∣ a ∣ is the amplitude, — 2π is the period. Master PHP 4 — the open source Web scripting breakthrough! Contains expert coverage of syntax, functions, design, and debugging! Leverage the amazing performance of the new Zend engine! 650+ real-world code examples! CD-ROM includes source code, plus everything you’ll need to run PHP 4 implementations on Windows and UNIX!. You can select the values of a and b within certain ranges. Sine wiggles in one dimension. We completed three cosine graphs together on the 13-5 vocab support page. Signal Processing. Real life word problems that involve the sine and cosine function can be used to keep the students engaged in the topic. ) Circles are an example of two sine waves. Find the amplitude and period of the function. Transformations of Trigonometric Functions. Graphs of the sine and cosine functions; and Periodic Behavior. Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. The student will graph sine or cosine graph affected by horizontal and vertical translations. Like What? [Filename: Notes 6. The Code of Federal Regulations is a codification of the general and permanent rules published in the Federal Register by the Executive departments and agencies of the Federal Government. It is given by parameter #a# in function #y = asinb(x - c) + d or y = acosb(x - c) + d# •The period of a graph is the distance on the x axis before the function repeats itself. • Use the cosine law to determine the interior angles of an acute triangle. The Sine Ratio Passy's World of Mathematics. Physical phenomenon such as tides, temperatures and amount of sunlight are all things that repeat themselves, and so are easily modeled by sine and cosine functions (collectively, they are called "sinusoidal functions"). Find an equation for a sine function that has amplitude of 4 a period of π. To understand how the absolute value can be applied to the real world, we'll look at two topics: What is considered normal with regard to the temperature of the human body and Fuel economy of two vehicles: a Honda Odyssey and a Nissan Altima. 8 L5 Trig Applications Part 1 - solve problems that arise from real world applications involving periodic phenomena D3. Lynn Weiss. Sketch the graphs of basic sine and cosine functions; use amplitude and period to help sketch the graphs of sine and cosine functions. Given an expression for the time rate of change of a quantity, find the initial rate. ppt), PDF File (. The maximum value appears to be 72 and the minimum value is 11. Goal: Students will use technology to discover how to graph trigonometric functions. Apr 15, 2018 - Discovering the graphs of Sine & Cosine using the measurements of the Unit Circle (Activity from Rice APSI) Stay safe and healthy. PreCalculus Honors - Curriculum Pacing Guide - 2017 - 2018 First Half of Semester Anderson School District Five Page 3 2017-2018 Unit 2 - Application of Trigonometry to Triangles (7 days) PC. solve problems involving trigonometry in acute triangles using the sine law and the cosine law, including problems arising from real-world applications; demonstrate an understanding of periodic relationships and the sine function, and make connections between the numeric, graphical, and algebraic representations of sine functions;. A sine wave is what you get when you plot the sine function on a graph. MCF 3M - Functions and Applications (Mixed) Spring 2014 Page history last edited by Peter Kim 5 years, 9 months ago This course introduces basic features of the function by extending students' experiences with quadratic relations. The product of two cosine functions is a sum of two other cosine functions. Trigonometric functions allow us to specify the shapes and proportions of objects independent of exact dimensions. Free practice questions for Precalculus - Graph the Sine or Cosine Function. Real world uses of hyperbolic trigonometric functions. Specific Expectations. Perform a sine regression using the Ti83/84 Activity #2 - Steamboat problem From given information: 1. Amplitude is first, then sine or cosine, then B, which you get by doing 2pi/10 (period), which reduces to pi/5, then it is the parenthesis with the transformation, and then last but not least is the sinusodial axis. 1) Find properties of sine functions (PC-N. Evaluating Inverse Trigonometric. Graphing Sine and Cosine Functions. \begingroup Basically, the answer is: yes, there are many other periodic functions, and the reason you typically see harmonics (like \sin,e^{i\omega t}) used is because in most simple applications of interest which are easily understandable, either the behavior itself is harmonic, or the behavior is most easily understood in terms of harmonics. First, you must understand that each of these functions has its own graph. In the chapter on Trigonometric Functions, we examined trigonometric functions such as the sine function. First of all, the graph is no longer a sine curve, but there's definitely a pattern to it. It exposes the student to the basic concepts of algebra and trigonometry with an emphasis in practical applications. But just saying "the sine function" doesn't tell you anything about the probability measure in the domain, which is what gets you probabilistic information, including joint distributions. the sine or cosine function and use the equation as a mathematical model to make predictions and interpretations about the real world. Using trigonometric functions to model climate. In ICSA, an elitism strategy is used to select the global solution, and a new updating mechanism for the new solution was proposed. You cannot graph an exponential functions without finding its horizontal asymptote. In addition, how do I know if this the graph of sine or cosine?. functions; increasing/decreasing functions; finding intercepts with axes. real world applications of sinusoids Sinusoids are applicable to the real world. To graph the sine function, we mark the angle along the horizontal x axis, and for each angle, we put the sine of that angle on the vertical y-axis. y-2π -π π 2π 0. The rise and fall of tides can have great impact on the communities and ecosystems that depend upon them. Every angle shares the same sine with another angle (and -sine with two others). • Find the amplitude and period of variation of the sine, cosine, and tangent functions. Sine and cosine functions are periodic functions. 4 COMPLEX NUMBERS Add and subtract complex numbers Multiply and divide complex numbers Simplify powers of i 2. The amplitude is 2. These graphs act as a reference every time you use a trigonometric function. Also for every n π/2 radians (where "n" is an interger) the cosine function passes through y=0. 21 from both sides: Divide both sides by 1. Outcome 5: Set up, solve, and graph equations from problems that require use of trigonometric functions, tangent, sine, and cosine and the Pythagorean Theorem. 3 make connections between the sine ratio and the sine function and between the cosine ratio and the cosine function by graphing the relationship between angles from 0º to 360º and the corresponding sine ratios or cosine ratios, defining this relationship as the function f(x) =sinx or f(x) =cosx, and explaining why the relationship is a function. The sine and cosine of an angle are both circular functions, and they are the fundamental circular functions. Full Range Fourier Series - various forms of the Fourier Series 3. Focus on creating a graph and formula from a story (starting from time 6:55) You Do Now: Warmup - Sketching a Cosine Wave. Learn how to construct trigonometric functions from their graphs or other features. Physical phenomenon such as tides, temperatures and amount of sunlight are all things that repeat themselves, and so are easily modeled by sine and cosine functions (collectively, they are called "sinusoidal functions"). 2-use technology, graphs, and tables to compare sine graphs. The property represented here is based on the right triangle and the two acute or complementary angles in a right triangle. The Code is divided into 50 titles which represent broad areas subject to Federal regulation. Sinusoidal Modeling - Real World Application Project. Then we looked at identifying the amplitude, frequency and period from a given cosine (or sine) function. Time-saving lesson video on Sine and Cosine Functions with clear explanations and tons of step-by-step examples. One way to distinguish sounds is to measure frequency. sinA sin B sin(AB) cosAcosB cos(AB) 59. 9) Graph sine and cosine functions (PC-N. 534), where x is the day of the year starting with January 1 as Day 1. Graph functions, plot data, evaluate equations, explore transformations, and much more - for free! Start Graphing Four Function and Scientific Check out the newest additions to the Desmos calculator family. Recall that the sine and cosine functions relate real number values to the x– and y-coordinates of a point on the unit circle. Day 7 HW F 15-Dec Graphing Cosine Functions NC. Determine the values for angles between 0° and 360°. Chapter Test. Evaluate and graph logistic growth functions. Find the amplitude and period of the function. Applications of Trigonometric Graphs; 6. Khan Academy is a 501(c)(3) nonprofit organization. Graph one period of s(x) = –cos(3x) The "minus" sign tells me that the graph is upside down. There are many uses of sin,cos,tan in real life. Trigonometric Functions • Understand some real-world situations that demonstrate periodic behavior. Write a cosine function that describes the height of the nail above ground as a function of angular distance. Trigonometry Graphing Trigonometric Functions Applications of Radian Measure. Discovery of Euler's Equation. Example 15 6. The new 3rd edition of Precalculus: Algebra and Trigonometry prepares students for further study in mathematics, the sciences, engineering, the social sciences, and other disciplines. By correctly labeling the coordinates of points A, B, and C, you will get the graph of the function given. 'transform') each of the three trigonometric functions discussed thus far: sine, cosine, and tangent. 25 scaffolded questions on equation graph involving amplitude and periodplus model problems explained step by step. Amplitude and Period of Sine and Cosine Functions BOATING A signal buoy between the coast of Hilton Head Island, South Carolina, and Savannah, Georgia, bobs up and down in a minor squall. Graphs of y = a sin x and y = a cos x 2. Yep, sine and cosine are practically twins. In maths, you have real life applications on any thing that you study. Lesson 7: Applications with Acute Triangles • Identify when to apply the sine and cosine laws given incomplete information about the • Solve a multistep problem that involves o two or more applications of the sine or cosine laws,. Sine Law and Cosine Law Find each measurement indicated. Free trigonometry worksheets, in PDF format, with solutions to download. Khan Academy is a 501(c)(3) nonprofit organization. stock market values can be formed into a sine, cosine graph (given a constant average) 5. Precalculus. Example: if your cutting down a tree and need to know where it will. I'm not exactly sure what level of mathematics you've studied so far, but sine and cosine are used heavily in math (calculus and functions) and physics (kinematics and dynamics), as well as having some real world applications. 2 The Inverse Trigonometric Functions. The trigonometric functions sine, cosine and tangent calculate the ratio of two sides in a right triangle when given an angle in that triangle. Although you. 4 The Real-Time Transport Protocol 12. the graph of a sine function. C2 create and analyse scatter plots of periodic data. Of course, this was to be expected, but this isn't the real problem; the real problem is that if your function has some crucial points it must pass through (which is certainly true for trigonometry functions), the truncation will move the curve away from those points. Standard 9: Find values of trigonometric functions using the unit circle. The unit circle and values of sine and cosine on the unit circle B. The fruit fly optimization algorithm (FOA) is a well-regarded algorithm for searching the global optimal solution by simulating the foraging behavior …. which means that theta can be any angle in degrees or radians — any real number. 6 u00ad Modeling Real World Data with Sinusoidal Functions Section 6. The unit circle can be used to express the values of sine, cosine, and tangent for π – x,. Students graph transformations of trigonometric functions (sine and cosine), including af(x), f(x) + d, f(x – c), and f(bx) for specific values of a, b, c, and d, in mathematical and real-world problem situations. Graphing Secant & Cosecant. Investigate Transformations of Sine and Cosine Functions Many real-world processes can be modelled with sinusoidal functions. The identities that arise from the triangle are called the cofunctionidentities. Sine and Cosine can be used in everyday life in various ways. Topic:Real world application of sine graphs. Press the button marked "sin" to change to the cosine. ratios, ie. Intro Tangent & Cotangent Graphs. To graph the sine function, we mark the angle along the horizontal x axis, and for each angle, we put the sine of that angle on the vertical y-axis. McGraw-Hill Education Trigonometry Review and Workbook, 1st Edition by William Clark and Sandra Luna McCune (9781260128925) Preview the textbook, purchase or get a FREE instructor-only desk copy. 3 make connections between the sine ratio and the sine function and between the cosine ratio and the cosine function by graphing the relationship between angles from 0º to 360º and the corresponding sine ratios or cosine ratios, defining this relationship as the function f(x) =sinx or f(x) =cosx, and explaining why the relationship is a function. If the set of input values is a ﬁnite set then the models are known as discrete models. org (pre-calculus), and Khan Academy (SAT. This builds into learning about graphing and interpreting logarithmic functions and models. Les't compare with. These graphs act as a reference every time you use a trigonometric function. 2) A car's tire has a diameter of 32 inches. The caternary curve (a dangling string/chain) is really just cosh – crasic Oct 30 '10 at 23:48. state the properties for the Sine and Cosine graph identify the values of a, k, c and d and apply it to a sinusoidal function to graph the resulting transformation Lesson:. 14-1 Graphs of Sine and Cosine Example 3: Sound Application Use a sine function to graph a sound wave with a period of 0. pdf), Text File (. A single note can be modeled on a sine curve, and a chord can be modeled with multiple sine curves used in conjunction with one another. Graphing Sine and Cosine Trig Functions With Transformations, Phase Shifts, Period - Domain & Range - Duration: 18:35. In the 36 intensively illustrated lectures of Mathematics Describing the Real World: Precalculus and Trigonometry, he takes you through all the major topics of a typical precalculus course taught in high school or college. The graphs of the sine and cosine functions illustrate a property that exists for several pairings of the different trig functions. Then you will learn about modeling trigonometric functions by graphing the sine and cosine functions. Recall from The Other Trigonometric Functions that we determined from the unit circle that the sine function is an odd function because [latex]sin(−x)=−sinx$. Sample records for nino-southern oscillation eventsnino-southern oscillation events «. The inverses of sine, cosine and tangent functions are not functions unless the domains are limited. Chapter Five – Trigonometric Functions What to Expect: A new formula designed for sin and cos graphs, details about frequencies, new vocabulary, explaining periodic trends, and the like. 2]Solve right and oblique triangles including application problems. Outcome 6: Set up and solve exponential and logarithmic equations; then identify and sketch graphs of the functions. Precalculus students are required to understand how sine and cosine functions model the real world. A lot of waves actually follow a sine graph, so we can prove that sinusoidal motion is a real thing in nature. We completed three cosine graphs together on the 13-5 vocab support page. f(x)= Calculus: Early Transcendental Functions Use Exercise 25 to find the moment of inertia of a circular disk of radius a with constant density about a. 0333 170 -170 t e 0. Have them graph these points and note whether or not they are on the same graph. Use a horizontal scale where one unit represents 0. With the sine law. Students will find equations for transformed sine curves and will graph transformed sine functions. Sketch translations of graphs of sine and cosine functions; use sine and cosine functions to model real-life data. Other circular functions can all be derived from the sine and cosine of an angle. 002 s and an amplitude of 3 cm. Examples: • Square wave function • Saw tooth functions Split Functions Much of the behaviour of current, charge and voltage in an AC circuit can be described. The inverses of sine, cosine and tangent functions are not functions unless the domains are limited. You will be observing the graph of r = a + bsin() or r = a + bcos(), where a and b are non-zero real numbers. The period of sine and cosine functions is 2pi/B. graph the sine and cosine functions with varied amplitudes and periods. Now we must stretch out the amplitude of the sine graph. Sketch the graphs of the sine and cosine functions and state their domain and range. Applications of Sine and Cosine Graphs Learning Task: Trigonometry functions are often used to model periodic data. The sum formula for tangent states that the tangent of the sum of two angles equals the sum of the tangents of the angles divided by 1 minus the product of the tangents of the angles. Graphs the 6 Trigonometric Functions. Rational functions: asymptotes and excluded values (PC-E. It is given by parameter #a# in function #y = asinb(x - c) + d or y = acosb(x - c) + d# •The period of a graph is the distance on the x axis before the function repeats itself. But just as you could make the basic quadratic, y = x 2, more complicated, such as y = -(x + 5) 2 - 3, so also trig graphs can be made more complicated. Trigonometry plays a major role in musical theory and production Real world examples of cosine functions. We typically use degree measures when measuring angles, however we can use radian angle measure as an alternate way of measuring angles in advanced math courses. KEY STANDARDS ADDRESSED: MA3A3. Many compression algorithms like JPEG use fourier transforms that rely on sin and cos. Trigonometric functions can be used to model real world data that is periodic in nature. Since the sine function y = sin t begins at the origin (when t = 0), sine is the more convenient of the two for this purpose. First, find two easily solvable angles that add up to 75. 1 The Inverse Sine, Cosine, and Tangent Functions. Objectives: To graph the sine and cosine functions; to identify the graphs of the sine and cosine functions. ' Or: sin(30 °) = 0. Domain and Range of Sine and Cosine The domain of sine and cosine is all real numbers, The range of sine and cosine is the interval [-1, 1] x or (, ) Both these graphs are considered sinusoidal graphs. Find Period and Amplitude 2. Not only is this a real-world visual representation of both graphs, but it also demonstrates that a cosine graph is simply a sine graph at a displacement of 90°. The table of values for the functions. Find the frequency in hertz for this sound wave. The graph of the sine function is called the sine curve. The property represented here is based on the right triangle and the two acute or complementary angles in a right triangle. 7 sketch graphs of y = af (k(x - d)) + c by applying one or more transformations to the graphs of f(x) =sinx and f(x) =cosx, and state the domain and range of the transformed functions 2. It is the reciprocal of the. Hyperbolic functions, also called hyperbolic trigonometric functions, the hyperbolic sine of z (written sinh z); the hyperbolic cosine of z (cosh z); the hyperbolic tangent of z (tanh z); and the hyperbolic cosecant, secant, and cotangent of z. Each of the six trig functions is equal to its co-function evaluated at the complementary angle. However, if we stretch the sine graph and change the amplitude, it just might work. In problems 12 & 13, the graphs of the sine and cosine functions are waveforms like the figure below. Since the multiplier out front is an "understood" –1, the amplitude is unchanged. If we change the number of cycles the wave completes every second -- in other words, if we change the period of the sine wave -- then we change the sound. Draw three right triangles and label the angle and two sides that apply to the sine, cosine and tangent functions respectively. Four Function Scientific. The hyperbolic functions take a real argument called a hyperbolic angle. Discovery of Euler's Equation. Cumulative Review. Lesson 9-11: Graphing Trigonometric Functions and Applications in Real World Contexts Learning Goals #8, 9, 10: How do I use the critical values of the Sine and Cosine curve to assist in in graphing Sine and Cosine functions? How do I contextualize Trig Functions in real life situations? Warm-Up! WITH A PARTNER. Water Depth Word Problem Modeled with Cosine Sine Function. One example of applying tangent functions to solve a real world problem is:- Find the gradient and the actual length of a path represented as x cm (a known measurable quantity) in the 1 : n (a known given quantity) scaled contour map. Sound waves travel in a repeating wave pattern, which can be represented graphically by sine and cosine functions. The SOH means that the sine of an angle equals the side opposite the angle over the triangle's hypotenuse. One way to distinguish sounds is to measure frequency. Animated gifs showing relationships between functions and geometry eg trigonometric functions How the unit circle gives us the sine wave Teaching math with the Drum Loop Real sinusoid on a timebase, formed by a linear increment of complex argument in time. Let’s look at a few examples of real-world situations that can best be modeled using trigonometric functions. This is an example of how a Graph of a Cosine Function looks like:. Use sinusoidal functions to solve problems. which means that theta can be any angle in degrees or radians — any real number. To understand how the absolute value can be applied to the real world, we’ll look at two topics: What is considered normal with regard to the temperature of the human body and; Fuel economy of two vehicles: a Honda Odyssey and a Nissan Altima. Periodicity of trig functions. Trigonometric identities like finding. Bookmark File PDF Functions Modeling Change Answers Functions Modeling Change Answers Math Help Fast (from someone who can actually explain it) See the real life story of how a cartoon dude got the better of math Modeling with Functions Part 1 We model real life scenarios of sales and volume of a box with functions. 4 Solving Word Problems Involving Sine or Cosine Functions. By finding a few key points or aspects of the graph, any of the real-life problems we have today can be explained mathematically and much of the vibrations surrounding us can be better understood. active oldest votes. 3π 2 = = Period Amplitude. Fortunately, we can analyze the problem by first representing it as a linear function and then interpreting the components of the function. Read: 'Zero point five is the sine of thirty degrees. Sound waves travel in a repeating wave pattern, which can be represented graphically by sine and cosine functions. Academic Precalculus. functions to model real world problems A cosine graph appears. Topic:Real world application of sine graphs. Trigonometric functions can be used to solve right triangles. Overview of Fourier Series - the definition of Fourier Series and how it is an example of a trigonometric infinite series 2. Maths Applications: Graph sketching. Euler's equation (formula) shows a deep relationship between the trigonometric function and complex exponential function.


# Sequences_and_Series - 11 Sequences and Series Consider the...

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Chapter 8 / Exercise 14
Essential Calculus: Early Transcendentals
Stewart
Expert Verified
11 Sequences and Series Consider the following sum: 1 2 + 1 4 + 1 8 + 1 16 + · · · + 1 2 i + · · · The dots at the end indicate that the sum goes on forever. Does this make sense? Can we assign a numerical value to an infinite sum? While at first it may seem di cult or impossible, we have certainly done something similar when we talked about one quantity getting “closer and closer” to a fixed quantity. Here we could ask whether, as we add more and more terms, the sum gets closer and closer to some fixed value. That is, look at 1 2 = 1 2 3 4 = 1 2 + 1 4 7 8 = 1 2 + 1 4 + 1 8 15 16 = 1 2 + 1 4 + 1 8 + 1 16 and so on, and ask whether these values have a limit. It seems pretty clear that they do, namely 1. In fact, as we will see, it’s not hard to show that 1 2 + 1 4 + 1 8 + 1 16 + · · · + 1 2 i = 2 i 1 2 i = 1 1 2 i 255
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Chapter 8 / Exercise 14
Essential Calculus: Early Transcendentals
Stewart
Expert Verified
256 Chapter 11 Sequences and Series and then lim i →∞ 1 1 2 i = 1 0 = 1 . There is one place that you have long accepted this notion of infinite sum without really thinking of it as a sum: 0 . 3333 ¯ 3 = 3 10 + 3 100 + 3 1000 + 3 10000 + · · · = 1 3 , for example, or 3 . 14159 . . . = 3 + 1 10 + 4 100 + 1 1000 + 5 10000 + 9 100000 + · · · = π . Our first task, then, to investigate infinite sums, called series , is to investigate limits of sequences of numbers. That is, we o cially call i =1 1 2 i = 1 2 + 1 4 + 1 8 + 1 16 + · · · + 1 2 i + · · · a series, while 1 2 , 3 4 , 7 8 , 15 16 , . . ., 2 i 1 2 i , . . . is a sequence, and i =1 1 2 i = lim i →∞ 2 i 1 2 i , that is, the value of a series is the limit of a particular sequence. While the idea of a sequence of numbers, a 1 , a 2 , a 3 , . . . is straightforward, it is useful to think of a sequence as a function. We have up until now dealt with functions whose domains are the real numbers, or a subset of the real numbers, like f ( x ) = sin x . A sequence is a function with domain the natural numbers N = { 1 , 2 , 3 , . . . } or the non-negative integers, Z 0 = { 0 , 1 , 2 , 3 , . . . } . The range of the function is still allowed to be the real numbers; in symbols, we say that a sequence is a function f : N R . Sequences are written in a few di ff erent ways, all equivalent; these all mean the same thing: a 1 , a 2 , a 3 , . . . { a n } n =1 { f ( n ) } n =1 As with functions on the real numbers, we will most often encounter sequences that can be expressed by a formula. We have already seen the sequence a i = f ( i ) = 1 1 / 2 i ,
11.1 Sequences 257 and others are easy to come by: f ( i ) = i i + 1 f ( n ) = 1 2 n f ( n ) = sin( n π / 6) f ( i ) = ( i 1)( i + 2) 2 i Frequently these formulas will make sense if thought of either as functions with domain R or N , though occasionally one will make sense only for integer values. Faced with a sequence we are interested in the limit lim i →∞ f ( i ) = lim i →∞ a i . We already understand lim x →∞ f ( x ) when x is a real valued variable; now we simply want to restrict the “input” values to be integers. No real di ff erence is required in the definition of limit, except that we specify, per- haps implicitly, that the variable is an integer. Compare this definition to definition 4.10.4.


# Finding the two planes that contain a given line and form the same angle with two other lines

I am asked the following question:

There are two planes $\pi_1$ and $\pi_2$ such that each of them contain the line $t \{ (-1,4,0) + \lambda (1,2,3)$ and form the same angle with the lines $r \{ (0,0,0) + \beta (1,1,1)$ and $s \{ (-1,2,3) + \alpha (1,1,-1)$. Find the angle formed between those planes

My solution:

Let's say the normal vector of $\pi$ is $\vec{n} = (a,b,c)$. Since the angle between the plane(s) and the lines are equal,

\begin{align*} \measuredangle \left( \pi, r \right) &= \measuredangle \left( \pi, s \right)\\ \frac{\left( a,b,c \right) \cdot (1,1,1)}{\sqrt{a^2+b^2+c^2} \ \sqrt{3}} &= \frac{\left( a,b,c \right) \cdot (1,1,-1)}{\sqrt{a^2+b^2+c^2} \ \sqrt{3}}\\ a+b+c &= a+b-c\\ c &= 0 \end{align*}

Also, since $t$ is contained in the plane, the dot product between its vector and the normal vector of the plane is zero:

\begin{align*} (a,b,c) \cdot (1,2,3) &= 0\\ a+2b+3c&=0\\ a &= -2b \end{align*}

Assuming that the normal vector has length one,

$$a^2 + b^2 + c^2 = 1$$

So we have three equations with three variables and the solution for those is

$$a = \mp \frac{2}{\sqrt{5}} \quad a = \pm \frac{1}{\sqrt{5}} \quad c = 0$$

so the normal vectors of the two planes found are

$$\left( \frac{2}{\sqrt{5}} , - \frac{1}{\sqrt{5}} , 0 \right) \quad \left( - \frac{2}{\sqrt{5}} , \frac{1}{\sqrt{5}} , 0 \right)$$

and the angle between them is

$$\theta = \arccos \left(\frac{\left \vert - \frac{2}{\sqrt{5}} \frac{2}{\sqrt{5}} - \frac{1}{\sqrt{5}} \frac{1}{\sqrt{5}} + 0 \right \vert}{1} \right) = \arccos(1) = 0^\circ$$

My answer does not agree with the textbook's solution.

Textbook's solution:

$$\theta = \arccos \left( \frac{9}{\sqrt{95}} \right)$$

Did I make a mistake somewhere?

Thank you.

$$|(a,b,c) \cdot (1,1,1)| = |(a,b,c) \cdot (1,1,-1)| \\ |a+b+c| = |a+b-c|$$ yields
$$a+b+c = a+b-c \quad \text{ and } \quad a+b+c = -a-b+c$$


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# A bag has only r red marbles and b blue marbles. If five red marbles

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A bag has only r red marbles and b blue marbles. If five red marbles  [#permalink]

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25 Jul 2018, 23:16
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A bag has only r red marbles and b blue marbles. If five red marbles and one blue marble are added to the bag, and if one marble is then selected at random from the bag, then the probability that the marble selected will be blue is represented by

A. b/r

B. b/(b + r)

C. (b + 1)/(r + 5)

D. (b + 1)/(r + b + 5)

E. (b + 1)/(r + b + 6)

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Re: A bag has only r red marbles and b blue marbles. If five red marbles  [#permalink]

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25 Jul 2018, 23:20
Number of red marbles = r

Number of blue marbles = b

five red marbles and one blue marble are added to the bag

Number of red marbles = r + 5

Number of blue marbles = b + 1

Total number of marbles = r + 5 + b + 1 = r + b + 6

Probability of selecting a blue marbles = $$\frac{b + 1}{r + b + 6}$$

Hence option E
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Re: A bag has only r red marbles and b blue marbles. If five red marbles  [#permalink]

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26 Jul 2018, 01:02
Bunuel wrote:
A bag has only r red marbles and b blue marbles. If five red marbles and one blue marble are added to the bag, and if one marble is then selected at random from the bag, then the probability that the marble selected will be blue is represented by

A. b/r

B. b/(b + r)

C. (b + 1)/(r + 5)

D. (b + 1)/(r + b + 5)

E. (b + 1)/(r + b + 6)

Red marbles = r

Blue marbles = b

Changing Situation :

Red = r + 5

Blue = b + 1

Total = r + s + 6

we are asked to pick up a blue marbles = $$\frac{( b + 1)}{r + s + 6}$$

Thus the best answer is E.
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Re: A bag has only r red marbles and b blue marbles. If five red marbles  [#permalink]

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30 Jul 2018, 10:51
Bunuel wrote:
A bag has only r red marbles and b blue marbles. If five red marbles and one blue marble are added to the bag, and if one marble is then selected at random from the bag, then the probability that the marble selected will be blue is represented by

A. b/r

B. b/(b + r)

C. (b + 1)/(r + 5)

D. (b + 1)/(r + b + 5)

E. (b + 1)/(r + b + 6)

After 5 red marbles and 1 blue marble are added to the bag, the bag has b + 1 blue marbles and r + b + 5 + 1 = r + b + 6 total marbles. So the probability of selecting a blue marble is (b + 1)/(b + r + 6).

_________________

Jeffery Miller

GMAT Quant Self-Study Course
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Re: A bag has only r red marbles and b blue marbles. If five red marbles &nbs [#permalink] 30 Jul 2018, 10:51
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