2 Educational Objectives After performing this experiment, students should be able to: 1. All quadratic functions fit the form y = Ax 2 + Bx + C, where A, B, and C can be any real number (although A cannot be zero). Trigonometry is a system that helps us to work out missing sides or angles in a triangle. Learning Targets: (1) I can sketch the graphs of the parent functions of sine and cosine. The sine and cosine functions are among the most important functions in all of mathematics. Among many uses and applications of the logistic function/hyperbolic tangent there are: which is based in the hyperbolic cosine function. Synthetic Geometry and Coordinate Geometry are used in real life to help us understand the dimensions and transformations of shapes and figures such as lines, triangles, polygons, and circles. Basic graph types are y = a sin(bx) + c and y = a cos(bx) + c. Some examples include the weather, seasonal sales of goods, body temperature, the tide's height in a harbor, average temperatures, and so on. • Table 1 gives the sum of two arbitrary cosine functions. Therefore, a time domain function f(x) and its corresponding frequency domain function are duals of each other. rewrite real-life formulas. 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[latex]\,\left[-1,1\right]. 03 Students will take the unit circle and construct the graphs of the sine and cosine functions. In this way, it has many engineering applications such as electronic circuits and mechanical engineering. θ = 90°), which gives us cos θ = 0 and. For example, variables that depend on the seasons may be modeled with trigonometric functions because the seasons repeat every year just like the sine function repeats every 2π. Here's an example: San Diego, California, is a gorgeous […]. Graph a sine or cosine function having a different amplitude and period. This equation is usually solved using sums of sines and cosines. Build an understanding of trigonometric functions by using tables, graphs and technology to represent the cosine and sine functions. Calculus is made up of Trigonometry and Algebra. distributions characterized by a cdf based on the sine function, called the new sine-G family of distributions. At 90° and 270°, x = 0 and therefore cos φ = 0, while at 0° and 180° y = 0 and therefore sin φ = 0. Explore symmetry (odd & even), domain, range, continuity, relative extrema and concavity of sine and cosine functions; Use Graphing Calc. The solution for an oblique triangle can be done with the application of the Law of Sine and Law of Cosine, simply called the Sine and Cosine Rules. These can be confusing to understand and memorize however these for the basis of all trigonometric values and derivations. Homework Statement High tide at 4am with a depth of 6 meters. Modeling with Sine or Cosine Functions Sine and cosine functions can be used to model real-world phenomena, such as sound waves. The midline is the average value. They are often shortened to sin, cos and tan. 4 Trigonometric Functions of Any Angle 4. C y KA sl ul d KrYirgMhlt os 4 3r 6e2s ke er Ivie kd F. Applications of Trigonometric and Circular Functions 281 CHAPTER OBJECTIVES t -FBSOUIFNFBOJOHTPGBNQMJUVEF QFSJPE QIBTFEJTQMBDFNFOU Applications of Trigonometric and Circular Functions CAS Suggestions argument of the cosine. The application problems show different ways that one could use sine and cosine funtions in real life. Answers to the do-it-yourself problems are on the "Other" page as well as other helpful websites. 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. What is Trigonometry? Trigonometry is a another form of math dealing with the relationship of the sides and angles of triangles. I struggle with finding. Physical: 23,. 7 - Applications of Sinusoidal Functions Applications of Sinusoidal Functions. This lesson will present real world examples that involve inverse trigonometric ratios. The trigonometric functions sine, cosine, and tangent are useful in a wide range of applications, from solving simple problems to advanced and complex problems. Over 90 new problems on differential equations and their applications. Understanding Calculus is a complete online introductory book that focuses on concepts. It studies the relationship between involving lengths and angles of triangle. Objectives: To graph the sine and cosine functions; to identify the graphs of the sine and cosine functions. In maths, you have real life applications on any thing that you study. At x = 0 degrees, sin x = 0 and cos x = 1. Transformations of Sine and. What are the Applications of Trigonometry in Real Life? Ans. (3) The student uses functions and their properties to model and solve real-life problems. Graphing Cosine Function Lesson Plan 1. Graph a sine or cosine function having a different amplitude and period. What are real-life applications of the sine and cosine wave applications? It's for a pre-calculus assignment and I have to make a presentation to the class about it. What Are Imaginary Numbers? are used in real-life applications, such as electricity, as well as quadratic equations. Recall from Graphs of the Sine and Cosine Functions that the period of the sine function and the cosine function is $$2π$$. Subhaschandra Singh, Department of Physics, Dhanamanjuri College of Science, Imphal – 795 001, Manipur, India. introduction. Law of Cosine [Just formula. Pg 439 #1-8, 33, 35, 37, 43-55 odd · Sketch the graphs of basic sine and cosine functions · Use amplitude and period to sketch the graphs of sine and cosine functions · Sketch translations of the graphs of sine and cosine functions. 3: Trigonometric Functions in Real life - Duration: 13:02. A Fourier series is a way of representing a periodic function as a (possibly infinite) sum of sine and cosine functions. S08 2 Learning Objectives Upon completing this module, you should be able to: 1. (b) If the power of cosine is odd (n=2k+1), save one cosine factor and use the identity sin 2 x + cos 2 x = 1 to convert the remaining factors in terms of sine. bc is the horizontal shift 4. This is exactly the reason why, when the graphs are plotted on grids as above, the cosine graph is equivalent to the sine graph, omitting the fact that it is shifted, or translated, 90° to the left. Practice Now! Trigonometric Ratios and Angle Measures Topics: 1. Tan x must be 0 (0 / 1) At x = 90 degrees, sin x = 1 and cos x = 0. Trigonometry plays a major role in musical theory and production. Outcome 5: Set up, solve, and graph equations from problems that require use of trigonometric functions, tangent, sine, and cosine and the Pythagorean Theorem. 4 9 Review Day 10 Test Day TOTAL DAYS: 10 C2. rewrite real-life formulas. Note the capital "C" in Cosine. Note the capital “C” in Cosine. A trigonometric function can be used to find the height of a smokestack on top of a building. Relationship between Sine and Cosine graphs The graph of sine has the same shape as the graph of cosine. &Assume&that&a& rider&enters&a&car&froma&platformthat&is&located&30°&around. Solitary Wave Solutions to Certain Nonlinear Evolution Equations by Rational Sine –Cosine Function Method S. Unit Five: Real World Problems Example: When Light shines through two narrow slits, a series of light and dark fringes appear. State the domain and range of sine, cosine, and tangent curves of the form y = Asinx, y = Acosx, and y = Atanx. Outcome 6: Set up and solve exponential and logarithmic equations; then identify and sketch graphs of the functions. F) as its Fourier transform. But such proofs are lengthy, too hard to reproduce when you’re in the middle of an exam or of some long calculation. 7m, but I am unsure how to interval the axis. Graphs of elementary trig functions allow you to see the graphs of sine, cosine and tangent and their relationship to travelling around a circle. Yes, you can derive them by strictly trigonometric means. Prerequisite Given the period, solve for the parameter '' in or. Trigonometry has many applications in astronomy, music, analysis of financial markets, and many more professions. Applications of the Derivative. This lesson will present real world examples that involve inverse trigonometric ratios. T3 and 20-2. The general form of a sinusoidal is: $f ( x ) = a \sin ( bx – c ) + d, \text{ for } b > 0$. So in the rule, c 2 = a + b - 2ab cos C side you are looking for angle opposite the side you want Label your diagram using a,b and c to avoid confusion Substitute into the formula and evaluate. Model the problem using the equation to show the depth of the water t hours after midnight. The slider is used for frequency values. Most radio communication is based on the use of combinations of sines and cosine waves. So our function is $y = 3. Objectives: To graph the sine and cosine functions; to identify the graphs of the sine and cosine functions. Experiment with functions that have additional terms, and see how these change the period, amplitude, and phase of the waves. Trigonometry plays a major role in musical theory and production Real world examples of cosine functions. What we want is the Law of Sines. Modeling Temperature Data Name(s): Since the trigonometric functions are periodic, they are a particularly useful tool when modeling cyclic behavior. They will also explore certain relationships that emerge from the measurement of three-dimensional figures and two-dimensional shapes. Especially sine and copsine functions have been very useful in applications for medical science, signal processing, geology, electronic communication, thermal analysis, satellite communication and many more. Find the period of a sine or cosine function. There are primarily three trigonometric functions commonly used with trigonometric identities to solve complex equations. The domain is the set of all real numbers. We explain Real World Examples of Inverse Trigonometry Functions with video tutorials and quizzes, using our Many Ways(TM) approach from multiple teachers. Trigonometric formulae are useful for solving problems in two dimensions. aphing Sine and Cosine functions: amplitude, phase shift, and vertical slide Cofunctions (Notes pp. Using Sum and Difference Formulas In this lesson, you will study formulas that allow you to evaluate trigonometric functions of the sum or difference of two angles.  cos^{-1}(x) = arccosx \rightarrow The arc that has a cosine of x. The Sine, Cosine and Tangent functions are often applied to real world scenarios. The applications of Fourier transform are abased on the following properties of Fourier transform. Real life scenario of logarithms is one of the most crucial concepts in our life. The Sine, Cosine and Tangent functions are often applied to real world scenarios. Real-life applications of. Objectives: To graph the sine and cosine functions; to identify the graphs of the sine and cosine functions. basic introduction into trigonometry. Real life application of trigonometry is use of trigonometry formula and its functions. Euler's formula. 3: Trigonometric Functions in Real life - Duration: 13:02. These trigonometric functions are extremely important in science, engineering and mathematics, and some familiarity with them will be assumed in most ﬁrst year university mathematics courses. 4 The Sine and Cosine Ratio Learning Goal: Determine the measures of the sides and angles in right triangles using the primary trigonometric ratios and the Pythagorean theorem; solve problems involving the measures of sides and angles in right triangles in real-life applications. com is a free math website that explains math in a simple way, and includes lots of examples, from Counting through Calculus. We say that this sinusoidal has a vertical shift of 1. The sine function has many real life applications, a few of which are: Triangulation, used in GPS-equipped cellphones, Musical notes, Submarine depth, Length of a zip. In addition, non-right angled triangles can be solved using the sine and cosine trigonometric functions. The domain of a linear function is all real numbers, and it's not possible to write every real number in a list. arbitrary sine function. There is only315 degrees of pizza left. Graphing Sine and Cosine Trig Functions With Transformations, Phase Shifts, Period - Domain & Range - Duration: 18:35. In this article, a sort of continuation, I will be discussing some applications of this formula. Change from roots to rational exponents. So the graph looks like a very simple wave. However, the basic sine function usually requires one or more transformations to fit the parameters of the process. Many other Fourier-related transforms have since been defined, extending the initial idea to other applications. Magnitude Amplitude of combined cosine and sine Phase Relative proportions of sine and cosine The Fourier Transform: Examples, Properties, Common Pairs Example: Fourier Transform of a Cosine f(t) = cos (2 st ) F (u ) = Z 1 1 f(t) e i2 ut dt = Z 1 1 cos (2 st ) e i2 ut dt = Z 1 1 cos (2 st ) [cos ( 2 ut ) + isin ( 2 ut )] dt = Z 1 1 cos (2 st. Such behavior occurs throughout nature and led to the discovery of rapidly rotating stars called pulsars in 1967. It just plots the sine function on the screen.  cos^{-1}(x) = arccosx \rightarrow The arc that has a cosine of x. The trigonometric functions sine and cosine have four important limit properties: You can use these properties to evaluate many limit problems involving the six basic trigonometric functions. A function may be defined by means of a power series. for the area of a triangle • use of sine rule and cosine rule for any triangle. This is how we find out “sine/cosine = tangent/1”. You want to figure out what the angle is of the bottom of your long chair to the ground. 28-1 Length of Arc. However, there are still disadvantages such as low solution accuracy and poor global search ability. In other words, for any value of $$x$$,. We show that, in some situations, the new sine-G models provide an interesting alternative to the sine-G models, with possible di erent targets in terms of modelling. Let's put our values in there: Now let's move some things around and get calculating: We're not done yet, though, we need to apply some inverse sine to both sides to get to B itself. And I'll leave those details to the. The function has the same domain, but a range of. General triangle word problems. Homework Statement High tide at 4am with a depth of 6 meters. Example 1: Evaluate. In Chapter XI of The Age of Reason, the American revolutionary and Enlightenment thinker Thomas Paine wrote:. In this way, it has many engineering applications such as electronic circuits and mechanical engineering. • Table 1 gives the sum of two arbitrary cosine functions. #91 - Sketch translations of graphs of sine and cosine functions. F MTH 123 Ginger Rohwer 28 February Real Life Trigonometry Applications In Lunar Phases In this project we will be collecting data about the illumination of the moon for certain months then creating sine and cosine functions to. We could think of the cosine function as a sine wave with phase shift ˇ 2. Law of Sines. Definitions and formulas for basic trigonometry, sine, cosine, tangent, cosecant, secant, cotangent, the law of sines and the law of cosines. The domain of the sine function is the set of real numbers, that is, every real number is a first element of one pair of the function. Besides other fields of mathematics, trig is used in physics, engineering, and chemistry. • Given information about a rotating functions in the real world. Together, these extensions define (sin φ, cos φ) for any angle φ, positive or negative, of any size. Plotting a basic sine wave. Relationship between Sine and Cosine graphs The graph of sine has the same shape as the graph of cosine. the sine, cosine and tangent functions are often applied to real world scenarios. The amplitude is a=2 and the period is. 3 Trigonometry in the Real World C2. You want to figure out what the angle is of the bottom of your long chair to the ground. SheLovesMath. Upon the completion of this course, you will have a better understanding of key geometry axioms and theorems. (3) I can sketch translations of the graphs of sine and cosine functions. Applications are given for four real data sets, showing a better t in comparison to some existing distributions based on some of goodness-of- t tests. The first portion of the book is an investigation of functions, exploring the graphical behavior of, interpretation of, and solutions to problems involving linear, polynomial, rational, exponential, and logarithmic functions. Sine, Cosine and tangent are the three important trigonometry ratios, based on which functions are defined. The amplitudes of the cosine waves are held in the variables: a1, a2, a3, a3, etc. 85 inches by 3. The graph of a quadratic function is called a parabola. Sine 5 None Up 2 9. Find an equation for a cosine function that has an amplitude of OTC — , a period of Find an equation for a sine function that has amplitude of 5, a period of 3TT. At x = 0 degrees, sin x = 0 and cos x = 1. This is how we find out “sine/cosine = tangent/1”. Possible Ideas Students will suggest:. Similar statements can be made for the other trigonometric functions of sums and differences. Synthetic Geometry and Coordinate Geometry are used in real life to help us understand the dimensions and transformations of shapes and figures such as lines, triangles, polygons, and circles. Mathematics Revision Guides - Real Life Trig Problems Page 12 of 14 Author: Mark Kudlowski Method 2 - Using the sine and cosine rules. Domain and Range of Sine and Cosine The domain of sine and cosine is all real. (Sine, Cosine, Secant, etc. The sine function takes an angle and tells the length of the y-component (rise) of that triangle. FIND the coordinates of A & C. The idea is to decompose any such function f(t) into an in nite sum, or series, of simpler functions. Hyperbolic functions show up in many real-life situations. We must now decide whether to use a sine function or a cosine function to get the phase shift. 11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e. Sine waves are very easy to produce electronically, and can be viewed on an oscilloscope, for. You decide to plug it into a sum and difference formula for sine. Use trigonometric functions to model and solve real-life problems. Graphs are shown in figure 4. SOH-CAH-TOA is a nice shortcut, but get a real understanding first! Gotcha: Remember Other Angles. It is analogous to a Taylor series, which represents functions as possibly infinite sums of monomial terms. 5b Translations of Sine and Cosine Curves Given the functions: y = a sin (bx – c) + d and y = a cos (bx – c) + d creates horizontal and vertical translations of the basic sine and cosine curves. The sine function is a set of ordered pairs of real numbers. In line 4 we use the properties of cosine (cos -x = cos x) and sine (sin -x = -sin x) to simplify the. Because angles are an intricate part of nature, sines, cosines and tangents are a few of the trigonometry functions ancient and modern architects use in their work. The period of tangent and cotangent are  \pi . Like all functions, trigonometric functions can be transformed by changing properties like the period, midline, and amplitude of the function. Find the equation of the normal to the curve of y=tan^-1(x/2) at x=3. We must now decide whether to use a sine function or a cosine function to get the phase shift. 4 8 More Real World Problems C2. Signal Processing. 3 Trigonometry in the Real World C2. Real life scenario of trigonometry is studied in cbse class 10. Trigonometric formulae are useful for solving problems in two dimensions. 7 Inverse Trigonometric Functions 4. Here are the topics that She Loves Math covers, as expanded below: Basic Math, Pre-Algebra, Beginning Algebra, Intermediate Algebra, Advanced Algebra, Pre-Calculus, Trigonometry, and Calculus. The Fourier series synthesis equation creates a continuous periodic signal with a fundamental frequency, f, by adding scaled cosine and sine waves with frequencies: f, 2 f, 3 f, 4 f, etc. In this article, a sort of continuation, I will be discussing some applications of this formula. Unit 4 - Graphing & Writing Sine & Cosine Functions; Application Problems October 21 to November 5th, 2013 Date Topic Assignment Monday Gr 10/21 changes. functions to find a efficient navigational route for long trips if GPS or other radar equipment is unavailable. The book has as much to do with calculus as with philosophy. Plotting more points gives the full shape of the sine and cosine functions. 4 9 Review Day 10 Test Day TOTAL DAYS: 10 C2. The mathematical topics of Fourier series and Fourier transforms rely heavily on knowledge of. 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 Pre-Cal 12 - Applications of Sinusoidal Functions New Project 5. FIND the coordinates of A & C. Part of the market-leading Graphing Approach Series by Larson, Hostetler, and Edwards, Precalculus Functions and Graphs: A Graphing Approach, 5/e, is an ideal student and instructor resource for courses that require the use of a graphing calculator. Substituting 0 for x, you find that cos x approaches 1 and sin x − 3 approaches −3; hence, Example 2: Evaluate. Definition 1 is the simplest and most intuitive definition of the sine and cosine function. A real life example of the sine function could be a ferris wheel. ) Circles are an example of two sine waves. The Fourier Series, the founding principle behind the eld of Fourier Analysis, is an in nite expansion of a function in terms of sines and cosines. 3 The symmetries of the six trig functions Since the sine function is odd and the cosine function is even then tan( ) = sin( ) cos( ) = sin( ) cos( ) = tan( ) and so the tangent function is odd. At 90° and 270°, x = 0 and therefore cos φ = 0, while at 0° and 180° y = 0 and therefore sin φ = 0. Return and go over Unit 2 Test. We have been studying how the equations of sinusoidal functions change when the graphs are shifted up and down and left and right and amplitude changes and period changes. In calculus, we are often. (See Example 3. The trigonometric functions sine and cosine have four important limit properties: You can use these properties to evaluate many limit problems involving the six basic trigonometric functions. GIVEN y x and B: 3cos2 0,3. The linear combination of a cosine and a sine of the same angle is an expression of the form: # a cos x + b sin x # That looks very much like the sum angle formula for sine or the difference angle formula for cosine:. Sine waves are a single note, whereas multiple sine waves make a chord. FREE (46) gnasher30 Uses of Trigonometry in Real Life. The book has as much to do with calculus as with philosophy. 3, P10Trig6. The PowerPoint is complete with custom animation so that pieces of the solving process are revealed, as well as the final answer. The Unit Circle is a tool used to find trigonometric functions. Basic Sine and Cosine Curves For 0≤ ≤2𝜋, the sine function has its maximum point at. (Check your answer with your. The tangent function differs in its shape and range from its sine and cosine counterparts. 4 Part 2 : Applications of Trigonometric Functions Self Evaluation Solutions Modelling Real Life Situations Worksheet with key Review Worksheet and Review Answers. (See Example 5. Decide whether you will need Pythagoras theorem, sine, cosine or tangent. The dynamic used in calculation fit with maximum precision and overflow is. This equation is usually solved using sums of sines and cosines. Sine, cosine, and tangent functions are used for many real life applications. Sine and cosine are keys to the success of Fourier Transform because sound may be. 42 on the original plot we get: Looks as though we are almost there. ) Sine or cosine functions can used to the motion. Many real world situations can be modeled using the sine and cosine functions. It may not have direct applications in solving practical issues but used in the various field. The heat equation is used to model how things get hot (electronics, spacecraft, ovens, etc). In maths, you have real life applications on any thing that you study. 28-3 Centroids. Web browsers do not support MATLAB commands. We used the law of sines again, sin(B)/(b)=sin(C)/(c) and fill in everything we know, reduced it down, solve for (b) and give our answer in terms of miles. 5 Objective: To draw (using technology), sketch, and analyse the graphs of sine and cosine function whose periods are rational numbers. In this way you can find the size of any unknown angle of a right triangle if you just know 2 sides of. Sine and cosine functions are used primarily in physics and engineering to model oscillatory behavior, such as the motion of a pendulum or the current In an AC circuit, but these functions also arise m other sciences in this project, you will consider an application to biology - we use sine functions to model the populations of a predator and its prey. B = sin-1 (0. We closed the day with an application of what the applet taught us about sine to help us graph cosine functions with translations. What is the value. 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 Pre-Cal 12 - Applications of Sinusoidal Functions New Project 5. Put the results in a table. The period of the function sin(x) is 2π. When the program returns, examine the stack for how many times the hyp sine was called and how many times hyp sine/cosine was called vs. For example, this transformed graph above would show which frequency sine and cosine functions to use to model our original function. Trigonometry Applications in Real Life. Applications of Trigonometric and Circular Functions 281 CHAPTER OBJECTIVES t -FBSOUIFNFBOJOHTPGBNQMJUVEF QFSJPE QIBTFEJTQMBDFNFOU Applications of Trigonometric and Circular Functions CAS Suggestions argument of the cosine. Stay safe and healthy. org are unblocked. The Sine Cosine Algorithms (SCA) has been recently proposed; it is a global optimization approach based on two trigonometric functions. The output of the transformation represents the image in the Fourier or frequency domain , while the input image is the spatial domain equivalent. In problems 12 & 13, the graphs of the sine and cosine functions are waveforms like the figure below. Area of a triangle 1. Hyperbolic functions show up in many real-life situations. & 3)&Theheight,ℎ,&in&meters,&of&the&tide&in&a&given&location&on&a&given&day&at&!&hours&after&midnight&can&be& modeled&using. I hope the following reasons and picture will help you understand why we study trigonometry. Describe that because they are functions, we need to be able to UNDO them (just like addition, subtraction, etc. In a right triangle, one angle is 90º and the side across from this angle is called the hypotenuse. m?X Law of Sines Substitute the given values. Experiment with functions that have additional terms, and see how these change the period, amplitude, and phase of the waves. Find and use reference angles to evaluate trigonometric functions. The hypotenuse is the longest side in a right triangle. The graphs of all sine and cosine functions are related to the graphs ofGOAL 2 Graph tangent y = sin x and y = cos xfunctions. Real-world Applications Using the Sine Function The table below shows the number of. Groupwork. The heat equation is used to model how things get hot (electronics, spacecraft, ovens, etc). Students will discover the area of triangles through the laws of sine and cosine. A1/2 ab sin c. Law of Cosine [Just formula. These can be confusing to understand and memorize however these for the basis of all trigonometric values and derivations. 1 Graphing Sine and Cosine Functions Focus on. Interpret the sine function as the relationship between the radian measure of an angle formed by the horizontal axis and a terminal ray on the unit circle and its y coordinate. PDF | In this paper, we propose a new hybrid algorithm called sine–cosine crow search algorithm that inherits advantages of two recently developed | Find, read and cite all the research you. Basic functions in TI-83 Graphing Calculator. Take my hyperbolic sin/cos recursive function place the angle on a sine or cosine stack that represents a call to the sine or cosine. " For example, "an oscilloscope is an electronic instrument used to display changing electrical signals. 0333 170 -170 t e 0. On a graph together, they look like this: Tangent The third basic trigonometric function is called the tangent (tan for short), and it is defined as the ratio of the opposite and adjacent sides - that is: tan θ = y. By correctly labeling the coordinates of points A, B, and C, you will get the graph of the function given. Solution of triangles. In Chapter XI of The Age of Reason, the American revolutionary and Enlightenment thinker Thomas Paine wrote:. Time-saving lesson video on Sine and Cosine Functions with clear explanations and tons of step-by-step examples. 3 The symmetries of the six trig functions Since the sine function is odd and the cosine function is even then tan( ) = sin( ) cos( ) = sin( ) cos( ) = tan( ) and so the tangent function is odd. Find the period of a sine or cosine function. 5); • From sine and cosine functions, we obtain tangent easily: sin tan cos θ θ θ =; (3). Plotting a basic sine wave. Nov 2: Graphs of sine and cosine; functions of the form f(x) = Asin(Bx+C); their amplitude and period. Find an equation for a sine function that has amplitude of 4, a period of fl. We can relate sides and angles in an arbitrary triangle using two basic formulas known as the sine rule and the cosine rule. Essentially, if what is being measured relies on a sine or cosine wave. All we must do now is stretch the period of the sine function. By Victor Powell. Practice: General triangle word problems. This standard works in conjunction with the content standards. Real Life Examples. Use sigma notation. This is a powerpoint explaining some of the applications of trigonometry to answer that age old question 'But Miss, why would we need to know this?&' Conte. The sinusoidal functions provide a good approximation for describing a circuit's input and output behavior not only in electrical engineering but in many branches of science and engineering. Explore the graph of general sine functions interactively using an HTML 5 applet. (Sine, Cosine, Secant, etc. This is in comparison to a continuous function like a line. It is the application of sinusoidal modeling that makes its use in the secondary school environment so valuable. If working outside, choose a spot with two widely spaced (2-5 meters) and roughly parallel lines to define the "river" banks. Trigonometric functions have wide variety of applications in real life. Who is the Father of Trigonometry? Ans. 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. In those tables, variables A and B are scalar constants, frequency ω is in radians/second, and variables α and β are phase angles measured. Thermal analysis. People board the ride at the ground (sinusoidal axis) and the highest and lowest heights you reach on the ride would be the amplitudes of the graph. 120 / 240 Vac sine Wave ac power distribution for residential application: The waveform of the electrical voltage distributed by the grid / the utility companies is like a sine wave. However, there are still disadvantages such as low solution accuracy and poor global search ability. introduction. Looks like MatPlotLib to me. Thefront panel of this instrument is 225 mm wide by 100 mm tall (8. However, in the real world all objects are three dimensional, so it is important that we extend the application of the area, sine and cosine formulae to three dimensional situations. The graphs of the sine and cosine functions are used to model wave motion and form the basis for applications ranging from tidal movement to signal processing which is fundamental in modern telecommunications and radio-astronomy. 2-5) Complete Notes pp. 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[latex]\,\left[-1,1\right]. Hyperbolic Functions in Real Life. The exponential function, exp(X) or e^X, is a special function that comes from calculus. More Graphing Trigonometric Functions Worksheet Answers Sec 5. Possible Ideas Students will suggest:. The cosine rule: c2 + b2 —2abcosC. 5 -Graphs of Sine and Cosine Functions What You'll Learn: #89 - Sketch the graphs of basic sine and cosine functions. Essentially, if what is being measured relies on a sine or cosine wave. 28-3 Centroids. And I'll leave those details to the. Specific Objectives (measurable) Use and manipulate visual representations of unit circle and trigonometric functions. and "use sine and cosine functions to model real-life data," iii. Some real life examples of periodic functions are the length of a day, voltage coming out of a wall socket and finding the depth of water at high or low tide. The general form of a sinusoidal is: \[ f ( x ) = a \sin ( bx – c ) + d, \text{ for } b > 0$. study investigates the performances of these estimates. Objectives: To graph the sine and cosine functions; to identify the graphs of the sine and cosine functions. On a graph together, they look like this: Tangent The third basic trigonometric function is called the tangent (tan for short), and it is defined as the ratio of the opposite and adjacent sides - that is: tan θ = y. This lesson will present real world examples that involve inverse trigonometric ratios. 4 Trigonometric Functions of Any Angle 4. If either the imaginary or the real part of the input function is zero, this will result in a symmetric Fourier transform just as the even/odd symmetry does. Sine, Cosine, Tangent Applications. , surveying problems, resultant forces). docx from MATH 313B at K12. They are both expressed according to the triangle on the right, where each letter. At the end of. c) How many minutes, from t = 0, does it take the rider to reach the. Recall that the cosine function takes an angle x as input and returns the cosine of that angle as output: For example if 60° is the input then 0. In these studies appear functions of the breast and cosine. Hyperbolic functions also satisfy identities analogous to those of the ordinary trigonometric functions and have important physical applications. 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OVERVIEW The students will learn how to interpret and graph an inverse trig. This is exactly the reason why, when the graphs are plotted on grids as above, the cosine graph is equivalent to the sine graph, omitting the fact that it is shifted, or translated, 90° to the left. Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosines, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real number. We should now understand that any variable that is cyclical, harmonic, oscillating, or periodic in nature can be modeled graphically by a sine or cosine wave. You and your friends order a pizza. At the end of. Learn how to graph trigonometric functions and how to interpret those graphs. Angle in standard position. If you can remember the graphs of the sine and cosine functions, you can use the identity above (that you need to learn anyway!) to make sure you get your asymptotes and x-intercepts in the right places when graphing the tangent function. c) How many minutes, from t = 0, does it take the rider to reach the. I know for sure that pilots use the trig. 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. Scroll down the page for examples and solutions. Sinusoidal functions graph wave forms. Date: 12/21/98 at 13:04:53 From: Doctor Santu Subject: Re: Trigonometry and music Dear Elizabeth, Certainly the functions sine and cosine have a connection to music. 2 to 5 Tuesday 10/22 Graphing Sine and Cosine functions cont'd. Applications of sinusoidal functions Description. Plotting a basic sine wave. Find the period of a sine or cosine function. View Notes - NOTESTrigonometry 3. 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In these trigonometry graphs, X-axis values of the angles are in radians, and on the y-axis its f (a), the value of the function at each given angle. The sine function just bounces back and forth (since you're basically tracing the outline of a circle). There is of course no point in simply copying this info here. The sine and cosine functions are among the most important functions in all of mathematics. "The sine and cosine functions are defined for all real numbers, and these functions have many real-world applications. y= cos 2x SOLUTION a. So I differentiate the sine function twice and I get, I claim minus the sine function. By thinking of sine and cosine as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval [−1,1]. We say that this sinusoidal has a vertical shift of 1. We then use the sine rule to find the side labelled a: sin12 40 sin16 a sin12. PPT TRIGONOMETRY in REAL LIFE - authorSTREAM Presentation. 0028 Open image in a new page Graph of e = 170 cos(120πt - π/3). The hypotenuse is the longest side in a right triangle. 2 to 5 Tuesday 10/22 Graphing Sine and Cosine functions cont'd. So that part, I hope, is clear. Just as with the sine function: • the domain for the cosine function is all real numbers; • the range of the cosine function is -1 ≤ y ≤ 1; • the cosine function is periodic and will repeat this pattern over intervals of 2π. The Sine Ratio Passy's World of Mathematics. • Apply addition or subtraction identities for sine, cosine, and tangent. I know for sure that pilots use the trig. The main goal is to illustrate how this theorem can be used to evaluate various types of integrals of real valued functions of real variable. In this article, a sort of continuation, I will be discussing some applications of this formula. Analysis of beams in mechanics (polynomial integration and optimization of a function on a closed interval) 2. Law of Sine. As you see, $$y = 1 + \sin x$$ merely raises the graph of sine one unit. The law of sines is a formula that helps you to find the measurement of a side or angle of any triangle. Double angle identities for sine and cosine. If it helps, you can draw a rough sketch to view this triangle, but this is optional. When we first learn the sine and cosine functions, we learn how to use them to find missing side-lengths & angles in right-angled triangles. The student is expected to: (A) use functions such as logarithmic, exponential, trigonometric, polynomial, etc. This would be a Fourier series with only one term, and would return the desired function with the magnitude changed. 2 to 5 Tuesday 10/22 Graphing Sine and Cosine functions cont'd. The sine and cosine graphs. In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. We check out triangle ABD and work out that angle ADB = 49°. and "use sine and cosine functions to model real-life data," iii. Applications of this branch of mathematics in real life are many and varied. A Fourier series is a way of representing a periodic function as a (possibly infinite) sum of sine and cosine functions. 4)AFerris&wheel&has&a&diameter&of&20&mand&is&4&mabove&ground&level&at&its&lowest&point. graphs of sine and cosine functions. 42 on the original plot we get: Looks as though we are almost there. 03 Real-world Applications of Trigonometric Functions. He is considered the father of trigonometry. Let's take a look at navigation. Peggy and Raymond then present groups with several real-world situations to work with. 5 –Graphs of Sine and Cosine Functions What You’ll Learn: #89 - Sketch the graphs of basic sine and cosine functions. (sine or cosine) inputs. 6ep7yrwj7rao, o7e5upezd1u3u, gw9joobvhirz2sm, w5yw1hcyy3, hgc3hy119dy, x0bluc6ksm4p, nztz99cmdv1t, ihnhyyxs5u9, f6mhu9u964bou, z52phaeu284nuz, ue99gp4bhf, 3x0oekbpvdy, jm2sczb2mnj1, lr41lj4kntz, bknyyhwsu4v, xlgusjsmamqhbf, olijtkg53n, n0oq7e9mjj, lordvnua01, xat1ajnfbhtu1j, ct63zp7jwogh, odkeljziwabr, 2otnx4k3rxnfck8, b6d986skqf, sbxpdkzn2rrj, 1hpc8sts6g2up, k4lj0jtibfrei, 19ltt17gps


# Solving the congruence $x^2 \equiv 4 \mod 105$. Is there an alternative to using Chinese Remainder Theorem multiple times?

I'm trying to solve $$x^2 \equiv 4 \mod 105.$$

This is of course equivalent to $$(x+2)(x-2) \equiv 0 \mod 105$$

which is also equivalent to the system of congruences

$$(x+2)(x-2) \equiv 0 \mod 3$$ $$(x+2)(x-2) \equiv 0 \mod 5$$ $$(x+2)(x-2) \equiv 0 \mod 7$$

which have solutions of $$x \equiv 1\ \text{or}\ 2$$ $$x \equiv 2\ \text{or}\ 3,\ \text{and}$$ $$x \equiv 2\ \text{or}\ 5$$ respectively.

Now, I could in principle take each combination of {$1,2$}, {$2,3$}, and {$2,5$} and use the Chinese Remainder Theorem to solve each system, but that seems incredibly tedious.

Is there a simpler way?

I'll note that this is a homework question, so it seems likely that there's a trick. Unfortunately I can't spot one.

• no trick....................................At the same time, some of your writing suggests you don't know what happens. What are the values $\pmod {15},$ just combine results for $3,5$ before going on – Will Jagy Feb 9 '17 at 21:14
• I do understand that I can just use the values that I obtain $\mod 15$ and then solve the system involving those values and the values $\mod 7$. Is that what you're getting at? – Alex Feb 9 '17 at 21:19
• sure. How many values $\pmod {15},$ and what are they? – Will Jagy Feb 9 '17 at 21:21

There are a couple ways to optimize. First you need only compute half of the $8$ combinations since if $\,x\equiv (a,b,c)\bmod (7,5,3)\,$ then $\,-x\equiv (-a,-b,-c)\bmod (7,5,3).\$ Then use CRT to solve it for general $\,a,b,c\,$ to get $\,x\equiv 15a+21b-35c\,$ Use that to compute those $4$ values. It's very easy

e.g. note $\ x\equiv (2,\color{#0a0}{-2},\color{#c00}2)\equiv 15(2)+21(\color{#0a0}{-2})-35(\color{#c00}2)\equiv 23\pmod{105}$

Negating it $\ (-2,\color{#0a0}2,\color{#c00}{-2})\equiv -x\equiv -23\equiv 82\pmod{105}.\$ Of course $\pm(2,2,2)\equiv \pm2.\$

Do as above for $\,(-2,2,2),\ (2,2,-2)\$ to get the other$\,4\,$ solutions (a couple minutes work).

Remark  There are also other ways we can exploit the negation symmetry on the solution space, i.e. if $x$ is a root so too is $-x$ since $\,x^2\equiv 2\,\Rightarrow\, (-x)^2\equiv x^2 \equiv 2\pmod{\!105}.\,$ Below is one such method, selected primarily because it reveals how to view Rob's answer in CRT language.

By CCRT = Constant case CRT: $\,x\equiv (2,2,2)\pmod{\!7,5,3}\iff x\equiv 2\pmod{\!105}.$ Its negation is $\,(-2,-2,-2)\,$ corresponding to $\,-2\pmod{\!105}.\,$ For other "nontrivial" solutions,  either $x$ or $-x$ has one entry $\equiv -2$ and both others $\equiv 2,\,$ say $\, x\equiv (-2,2,2)\pmod{p,q,r}.\,$ Again by CCRT $\,x\equiv (2,2)\pmod{q,r}\iff x\equiv 2\pmod{qr}$ by $\,q,r\,$ coprime. So we reduce from $3$ to the $2$ congruences below. Solving them by Easy CRT, using $p$ coprime to $qr,\,$ we get

\quad\ \ \begin{align} x&\equiv -2\!\pmod p\\ x&\equiv\ \ \,2\!\pmod{qr}\end{align}\!\iff x\equiv 2\ +\ qr\left[\,\dfrac{-4}{qr}\ \bmod\ p\,\right]\pmod{pqr}\,

\qquad \qquad\qquad\qquad\! \begin{align} p=7\,\ \Rightarrow\,\ x &\equiv 2 + 3\cdot 5(-4/(\color{#c00}{3\cdot 5)})\bmod 7)\equiv 2+3\cdot 5(3)\equiv 47\\[.2em] p=5\,\ \Rightarrow\,\ x&\equiv 2 + 3\cdot 7(-4/(\color{#c00}{3\cdot 7}))\bmod 5)\equiv 2+3\cdot 7(1)\equiv 23\\[.2em] p=3\,\ \Rightarrow\,\ x&\equiv 2 + 5\cdot 7{(-4/\!\!\!\underbrace{(\color{#0a0}{5\cdot 7})}_{\large \equiv\ \color{#c00}{1}\ {\rm or}\ \color{#0a0}{-1}}}\!\!\!)\bmod 3)\equiv 2\color{}{ +}5\cdot 7(1)\equiv 37\\ \end{align}

We arranged the above to exploit easy inverses  (of $\,\color{#c00}{1}$ or $\color{#0a0}{-1})\,$ just as in the first solution (cf. my comment below). So $\,47,23,37\,$ and their negatives $\,58,82,68\,$ are all the nontrivial solutions.

The method in Rob's answer is essentially equivalent to the above (without the CRT language), except it doesn't take advantage of the easy inverses, instead solving the congruences by brute force (sometimes this may be quicker than general methods when the numbers are small enough).

There is also another CRT optimization used implicitly in Rob's answer. Namely a change of variables $\ y = x\!-\!2\,$ is performed to shift one of the congruences into the form $\,y\equiv 0,\,$ which makes it easy to eliminate explicit use of CRT. We show how this works for the prior congruences, using the mod Distributive Law $\,ca\bmod cn =\, c(a\bmod n)\quad\qquad$

$\qquad qr\mid x\!-\!2\,\ \Rightarrow\,\ x\!-\!2\bmod{pqr}\, =\, qr\!\!\!\!\!\!\overbrace{\left[\dfrac{x\!-\!2}{qr}\bmod p\right]}^{\large\quad\ \ x\ \equiv\ -2\pmod{p}\ \ \Rightarrow}\!\!\!\!\!\!\! =\, qr\left[\dfrac{-4}{qr}\bmod p\right]$

That's the same solution for $\,x\!-\!2\,$ that Easy CRT gave above. So the mod Distributive Law provides a "shifty" way to apply CRT in operational form - one that often proves handy.

• Thanks. This seems like a bit more reasonable of an approach than just explicitly using CRT $8$ times. It's not entirely clear to me why we have $x \equiv 15a+21b-35c$. I recognize that those are the three different ways we can multiply {$3,5,7$}, but why? – Alex Feb 10 '17 at 14:51
• @Alex It follows by applying the general CRT formula, namely \begin{align} x\ &\equiv\ (a,b,c)\!\!\pmod{7,5,3}\\[.3em] \overset{\rm CRT}\iff x\ &\equiv\ a(3\cdot 5)\left[\dfrac{1}{3\cdot 5}\right]_7 + b(3\cdot 7)\left[\dfrac{1}{3\cdot 7}\right]_5 + c(5\cdot 7) \left[\dfrac{1}{5\cdot 7}\right]_3 \pmod{105}\\[.3em] &\equiv \quad a\,(15)\,(1/1)_7\ \ +\ \ b\,(21)\,(1/1)_5\ \ +\ \ c\,(35)\,(1/(-1))_3\\[.3em] &\equiv\ \ 15\,a\ +\ 21\,b\ -\ 35\,c \end{align} Note that the problem was designed to make this very easy, i.e. all the inverses are inverses of $\pm1.$ $\quad$ – Bill Dubuque Feb 10 '17 at 15:42
• What you describe as a computation by "brute force" in my answer amounts to a quick pass over a list of $12$ natural numbers $n$ less than $100$ to decide whether $n\pm4$ is a multiple of one of $3$, $5$, $7$. You seem to be doing much more calculation than I did. – Rob Arthan Feb 10 '17 at 23:05
• @Rob First, "brute force" is a very common name for such exhaustive searches. It's not meant to denigrate such methods (which are often useful). Second, if you show all your work (as I do) then it will be of comparable length (if not more). Third, the above is efficient for any size numbers, but brute force searching is not. Finally, above I do not aim to optimize brevity, Rather, I aim to highlight the essence of the matter, from a more general viewpoint. – Bill Dubuque Feb 10 '17 at 23:14
• @BillDubuque: first: the essence of this problem to me is "how can a quadratic equation have more than two roots in a commutative ring?". Second: tabulating the quick pass over a list of 12 small natural numbers is much quicker than any application of the CRT formula. – Rob Arthan Feb 10 '17 at 23:26

Here's a method for avoiding CRT altogether (which comes from thinking about why there can be more than two roots to a quadratic in a commutative ring). If $x$ is any solution, we have $$(x - 2)(x + 2) \equiv 0 \bmod 105.$$ So in addition to the easy solutions $x = 2$ and $x = 103$, we get a solution $x = y + 2$ for any $y$ such that $$y(y + 4) \equiv 0 \bmod 105.$$ I.e., for any $y$ such that $y$ and $y + 4$ are a "complementary" pair of zero divisors in the ring $\Bbb{Z}_{105}$. Any such pair has one of the following forms (possibly with the order of the factors reversed). \begin{align*} 3m &\cdot 35n & \quad &\mbox{with 1 \le n \le 2}\\ 15m &\cdot 7n & \quad &\mbox{with 1 \le m \le 6}\\ 5m &\cdot 21n & \quad &\mbox{with 1 \le n \le 4} \end{align*} It doesn't take long to work through the $2+6+4$ possibilities for $y$ or $y + 4$ to get that $y(y+4)$ must be one of:

$$21 \cdot 25\\ 35 \cdot 39 \\ 45 \cdot 49\\ 56 \cdot 60\\ 66 \cdot 70\\ 80 \cdot 84$$

Giving 23, 37, 47, 58, 68 and 82 as the not-so-easy solutions.

• As someone currently taking a ring theory class, I think this solution is really neat. Could you clarify why we have bounds on $n$ and $m$ in the third chunk of centred math? I don't exactly follow why a "complementary pair of zero divisors" must have one of those forms. It makes sense to me that you can split up the prime factors $3,5,7$ but I can't figure out why the $n$ and $m$ are necessary, and how they're chosen. – Alex Feb 9 '17 at 22:56
• @Alex: I think you understand that to get a non-trivial solution to $ab=0$ in $\Bbb{Z}_{105}$, you need to split the prime factors of $105$ between $a$ and $b$. Then you can multiply either $a$ or $b$ by any number you like, but as the equivalence class mod $105$ is all that matters, you only need to look at multiples between $0$ and $104$. My case analysis just picks the bigger of $a$ and $b$ to give the smallest number of cases. Then you just test whether the resulting multiple of $a$ (or $b$) $\pm4$ is a multiple of the "other" factor. – Rob Arthan Feb 9 '17 at 23:09
• @Alex Though CRT is not explicitly used above, in fact the method is a special case of general CRT optimizations, as I describe in the Remark in my answer. If one does many manual CRT calculations it proves handy to be familiar with these and related ideas. – Bill Dubuque Feb 10 '17 at 22:32


# Selecting $2$ cards from a full deck of cards.

I thought of a problem earlier and I am quite clueless on how to solve it, or begin solving it, because I cannot find a way to easily compute the amount of combinations of $$2$$ cards that sum to a certain value $$x$$. Anyway, the first part of the problem is as follows

We have a full deck of $$52$$ cards, and randomly select $$2$$ cards from this deck. We look at the cards and compute the sum of the values of the cards, ace being $$1$$ and K, Q and J being $$13$$, $$12$$ and $$11$$ respectively. We then shuffle the cards back into the deck and randomly select $$2$$ cards once more. What is the probability that the sum of the value of these $$2$$ cards, is the same as the sum of the values of the first $$2$$ selected cards?

This brought me to think of another problem, which is comparable. It is as follows

Let's say we have a full deck of $$52$$ cards, we randomly select $$2$$ cards, and we do this twice, yielding $$2$$ sets of $$2$$ cards. What is the probability that the sum of the values of the cards in the first set, equals the sum of the values of the cards in the second set?

Again, I'm quite confused about this problem, because I cannot think of an easy way to compute the amount of possible configurations of two cards, that sum to some value $$x$$.

Any help on solving these problems is appreciated. Furthermore, what would be a good guesstimate for these probabilities that could be given without any computations?

• Have you tried to work out the number of combinations for any specific sums? How many combinations add up to $2?$ What about $3?$ Try a few examples and look for pattern. – saulspatz Feb 17 at 17:34
• I'm thinking a decent guesstimate is in the area of $1/338 = 1/(2 \cdot 13^2)$. – Robert Shore Feb 17 at 17:35
• @robjohn As a guesstimate or as a result of a computation? – S. Crim Feb 17 at 18:29
• @S.Crim: It was a computation, but I had a replacement in there that shouldn't have been, so I need to compute again. – robjohn Feb 17 at 18:35
• @saulspatz I've managed to solve the first problem, but am now stuck on the second problem because it is confusing to me what happens when the sum is unknown. – S. Crim Feb 17 at 18:36

For an interview, I'd try to compare the second scenario to the first.

Say that you draw $$6$$ and $$3$$ the first time. If you drew from a separate deck, you'd have $$16$$ ways to make $$9,$$ but if you draw from the same deck, you only have $$9$$ ways. In the worst case you draw two aces, or two Kings, and the number of ways to succeed goes down from $$6$$ to $$1.$$ In the best case, you draw two $$7$$s and the number of ways goes down from 96 to 91. $$102$$ to $$97$$.

Of course, the better cases, near the middle will occur more frequently than the worse cases, near the ends, so I would guess that the probability of success in the second case might be about $$\frac34$$ of the probability of success in the first case.

Drat! Now I'm going to have to work them out and see how close I came.

EDIT

In response to the OP's that we don't have an explicit answer for the first problem either, in an interview, I would do the same kind of estimation. There are $${52\choose2}={52\cdot51\over2}\approx 1300$$ ways to chose two cards. For a sum of $$2$$ we must choose two Aces, and there are only $$6$$ ways to do that, so about half a percent chance of success. If the sum is $$14,$$ we have $$102$$ chances of success, so around $$8\%.$$ Of course, the sums near $$14$$ will arise more frequently than the sums near $$2$$ or $$26$$ so they should be weighted more heavily. I'd guess around $$5$$ percent.

With one deck, the probability of success is $${328888\over6497400}\approx .05062$$ and with two decks, the probability of success is $${365392\over7033104}\approx.05195$$ I computed the exact probabilities with a script, and confirmed them through simulation.

• Hah! That sounds good. However, for the first scenario, we cannot give an explicit probability, e.g. 60%, right? Because our probability depends on the outcome of the first sum, as you stated. So then how exactly would we compare the two situations in terms of probability of getting equal sums? – S. Crim Feb 17 at 20:30
• @S.Crim I was assuming that assuming that you'd already somehow come up with answer for the first scenario. In one of your comments, you said "I've managed to solve the first problem." I'll add some more to my answer. – saulspatz Feb 17 at 21:38
• Yeah I solved the first problem to the point where, knowing the outcome of the sum, I can compute the probability of drawing this sum again after reshuffling the drawn pair back in. So I cannot compute some explicit formula or something for the probability yet. – S. Crim Feb 17 at 21:56

Hint:

$$\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|} \hline + & \color{red}{1} & \color{red}{2} & \color{red}{3} & \color{red}{4} & \color{red}{5} & \color{red}{6} & \color{red}{7} & \color{red}{8} & \color{red}{9} & \color{red}{10} & \color{red}{11} & \color{red}{12} & \color{red}{13} \\ \hline \color{red}{1} & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 \\ \hline \color{red}{2} & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline \color{red}{3} & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ \hline \color{red}{4} & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 \\ \hline \color{red}{5} & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 \\ \hline \color{red}{6} & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 \\ \hline \color{red}{7} & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\ \hline \color{red}{8} & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 \\ \hline \color{red}{9} & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 \\ \hline \color{red}{10} & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 \\ \hline \color{red}{11} & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 \\ \hline \color{red}{12} & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 \\ \hline \color{red}{13} & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 & 26 \\ \hline \end{array}$$

The row of red numbers is the possible values for the 1st card, the column of red numbers is the possible values for the 2nd card. How many ways are there to get, say, a total sum of $$9$$?

• I see, very helpful. This is probably a good tool for me to try and find the solution. Thanks alot! – S. Crim Feb 17 at 18:14
• This allowed me to find the solution to first scenario, but I'm still stuck on the second one. Because if we do not know the value of the sum yet, then how can we compute the probability that the two sums are equal? I imagine we again have to look at possibilities or amounts of configurations but I'm not sure. – S. Crim Feb 17 at 18:20
• Use the Law of Total Probability. – Graham Kemp Feb 18 at 22:46

On the problem you thought about

We select cards $$c_1, c_2, c_3, c_4$$ from the same deck; there are $$U:=52\cdot 51\cdot 50\cdot 49= 6497400$$ possible cases, and we wish to count the cases for which $$N(c_1)+N(c_2)=N(c_3)+N(c_4) \label{eq1}\tag{1}.$$

Let $$V$$ denote the count of cases that satisfy $$(\ref{eq1})$$, and let us start by figuring out how many cases there are that satisfy $$N(c_1)+N(c_2) =n= N(c_3)+N(c_4),\label{eq2}\tag{2}$$ for some fixed $$n$$:

As robjohn says, there are $$W(n)=208-16\,|n-14|-4\,[\,2\mid n\,]$$ ways we can select $$c_1, c_2$$ so that they sum to $$n$$. (The formula for $$W(n)$$ should be clear from the square provided by cansomeonehelpmeout.) Fix such a choice of $$c_1, c_2$$; if we want to continue selecting $$c_3, c_4$$ such that they sum to $$n$$ too, then they must be chosen from the remaining ways $$W(n)-Q,$$ where $$Q$$ is the number of ways to select two cards that sum to $$n$$ such that at least one of the two cards is either $$c_1$$ or $$c_2$$. When $$N(c_1)\neq N(c_2)$$ (which is always true if $$n$$ is odd), contemplating on some specific cases, it does not take much effort to realize that $$Q$$ is always $$14$$. On the other hand, if $$n$$ is even, of all the choices we could have made for $$c_1, c_2$$, exactly $$12$$ of them have $$N(c_1)=N(c_2)$$, and for each one of these we have $$Q=10$$ and not $$14$$. Therefore, we let $$Q=14$$ and we add $$(14-10)\cdot12$$ to the total when $$n$$ is even to compensate for the extras we have subtracted, and we get the number of cases satisfying $$(\ref{eq2})$$ as $$W(n)(W(n)-14)+4\cdot 12\,[\,2\mid n\,]. \label{eq3}\tag{3}$$

All we need to do now is to sum $$(\ref{eq3})$$ over all possible $$n$$: $$V=\sum_{n=2}^{26}\left(W(n)(W(n)-14)+4\cdot 12\,[\,2\mid n\,]\right).$$ The probability that we get $$(\ref{eq1})$$ will then be: $$\frac{V}{U}.$$

Note: The computation by hand is time consuming. I would use a program to calculate the sum and give a final answer, but the software I use is a very old version of Maple, and it is very cumbersome.

Ways to draw $$n$$ with $$2$$ cards: $$W(n)=\overbrace{208-16\,|n-14|}^{16\,(13-|n-14|)}-4\,[\,2\mid n\,]$$ That is, there are $$13-|n-14|$$ ways to choose $$1\le a,b\le13$$ so that $$a+b=n$$. There are $$4\cdot4=16$$ card choices for $$a$$ and $$b$$, except if $$2\mid n$$ and $$a=b$$, when there are only $$4\cdot3=12$$, so we subtract $$4$$ if $$2\mid n$$. $$\sum_{n=2}^{26}W(n)=2652$$ The probability of drawing $$n$$ is then $$\frac{W(n)}{2652}$$ The probability of drawing the same thing twice in a row, replacing between draws, is then $$\frac1{2652^2}\sum_{n=2}^{26}W(n)^2=\frac{22837}{439569}\doteq0.051953$$

• Thanks alot for the answer. Could you elaborate a little more on the formule you gave for $W(n)$? What does the $208$ mean, and what exactly does it mean that something only occurs when $2|n$? Perhaps I am just unfamiliar with notations. – S. Crim Feb 18 at 15:08
• @S.Crim: I have hopefully made the explanation better. – robjohn Feb 19 at 8:05
• Thanks alot, that indeed clarifies some things. – S. Crim Feb 19 at 14:45
• @S.Crim: Is something still unclear? $[\dots]$ are Iverson Brackets and $a\mid b$ means that $a$ divides $b$; e.g. $[\,2\mid n\,]$ is $1$ when $n$ is even and $0$ when $n$ is odd. – robjohn Feb 19 at 15:05
• I think I get it now, yeah. Thanks for the clarifications on the notation. I was unfamiliar with the Iverson Brackets before this, and its been a while since I've worked with the $a|b$ notation. Thanks alot! – S. Crim Feb 19 at 17:45