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Chapter 6 Extending Periodic Functions Chapter Outline Section 6.1 You will solve trig equations by looking at a graph, the unit circle, and using the inverse functions on your calculator. You will see how several trig equations and applications have more than one solution. You will also investigate the inverse trig functions. Section 6.2 You will be extending your modeling of periodic functions to more complex situations. You will see how a trig function with two different angles can be simplified. Section 6.3 You will work with other trig formulas and identities to solve more complex trig equations. Section 6.4 You will see how the motion of a spring can be modeled by using a combination of a sinusoidal and an exponential function. Math Notes: Solving Periodic Functions When solving periodic functions (such as π ππ π₯ = β ), we get an infinite number of solutions (or no solution). In the example π ππ π₯ = β , we get the solutions: π₯= β¦β , , β¦ and π₯ = β¦ β , +2ππ, This may be written more compactly as x = , , ,β¦ +2ππ, where π is any integer. If we restrict the domain to [0, 2π), we get only two solutions: π₯ = and . Math Notes: Notation for Inverse Trigonometry There are two standard notations to represent the inverse for trig functions. The first uses inverse notation. For example: π¦ = π ππ β1 π₯ πππ π¦ = πππ β1 π₯ You must be careful with this notation. The function π¦ = π ππ β1 π₯ is not the same function as π¦= 1 π ππ π₯ . The function, sin x, while the function π¦ = π ππ β1 π¦= 1 π ππ π₯ is π¦ = ππ π π₯, the reciprocal function for π₯ is called the inverse function for π ππ π₯. Another form is using π¦ = ππππ ππ π₯ for the inverse of sine and π¦ = ππππππ π₯ for the inverse of cosine. This notation, π¦ = ππππ ππ π₯ can be thought of as βthe arc whose sine is β¦β This connects the inverse back to the unit circle. Both π¦ = π ππ β1 π₯ and π¦ = ππππ ππ π₯ mean exactly the same thing. π¦ = πππ β1 π₯ and π¦ = ππππππ π₯ are two different ways to write the exact same thing. Math Notes: Solving SSA When solving a triangle with two sides and a non-included angle (SSA), you need to be aware that the triangle can have no solution, one solution, or two solutions. If you use the Law of Sines to solve this type of triangle, you will need to find the missing angle first. Case 1 Here, side a is too short to make a triangle; no matter how we swing it from B, it wonβt reach the other side. Case 2 Here, side a is too long. When we swing it from B, it hits the opposite side in two places: Cand Cβ². ΞABC and ΞABCβ² both have a 30° angle and sides of 10 and a. Case 3 Here, side a is just right. When we swing it from B, it touches the opposite side at just one point, called C. There will be just one triangle, ΞABC. Math Notes: Angle of Inclination The tangent of the angle of inclination is the slope of a line. So the line y = mx + b makes an angle ΞΈ with the xaxis where tan ΞΈ = m. Note that ΞΈ is always acute. If the slope of the line is negative, the angle ΞΈ will be negative. Math Notes: 5-Point Graphing Method Reminder When graphing a sine or cosine function, it helps to locate 5 key points that describe the graph. These points are shown on the graph as black dots. π For example: π¦ = 2πππ ( (π₯ β 2)) + 1. 3 The ππππππ‘π’ππ = 2. The graph is shifted up 1 and right 2. Use the amplitude and the shifts to determine the first point. π Angular frequency is , so the period is found by 3 π π β = 2π, so π = 6. 3 Divide the period into 4 equal pieces to determine the spacing between points. Plot the remaining four points by moving to the middle, then bottom, back to the middle and finally back to the top. Math Notes: General Equation for Cosine If π¦ = πππ π₯ is stretched vertically by a, has an angular frequency of b, is shifted h units to the right, and k units up, the resulting graph has the equation: π¦ = π πππ (π(π₯ β β)) + π. The amplitude is a . The period (p) and angular frequency (b) are related by π · π = 2π. Math Notes: Sum and Difference Formulas Although sine and cosine do not distribute over sums or differences, it turns out that sin(Ξ± ± Ξ²) and cos(Ξ± ± Ξ²) can be expressed in terms of sin Ξ±, cos Ξ±, sin Ξ², and cos Ξ². In this and future math courses, you will need to use these identities many times to rewrite sin(Ξ± ± Ξ²) and cos(Ξ± ± Ξ²). Angle Sum Identities sin(Ξ± + Ξ²) = sin Ξ± cos Ξ² + cos Ξ± sin Ξ² cos(Ξ± + Ξ²) = cos Ξ± cos Ξ² β sin Ξ± sin Ξ² Angle Difference Identities sin(Ξ± β Ξ²) = sin Ξ± cos Ξ² β cos Ξ± sin Ξ² cos(Ξ± β Ξ²) = cos Ξ± cos Ξ² + sin Ξ± sin Ξ² Combined Forms sin(Ξ± ± Ξ²) = sin Ξ± cos Ξ² ± cos Ξ± sin Ξ² cos(Ξ± ± Ξ²) = cos Ξ± cos Ξ² β sin Ξ± sin Ξ² Math Notes: Double-Angle Formulas for Sine and Cosine cos 2Ξ± = cos2 Ξ± β sin2 Ξ± OR = 2cos2 Ξ± β 1 OR = 1 β 2 sin2 Ξ± sin 2Ξ± = 2 sin Ξ± cos Ξ± Math Notes: Half-Angle Formulas for Sine and Cosine The half-angle formulas are listed below. You must determine which sign to use ( + or β) from the π quadrant in which lies. 2 Also useful, are the formulas you found as preliminary steps to the half angle formula: and Chapter 6 Closure Problems with Answers Included for Self-Checking CL 6-162. Explain why π ππ π₯ 1 = 2 has more than one solution. a. Use the unit circle to justify your answer. b. Use a graph to justify your answer. c. What does π ππ β1 1 1 (2) equal? How is this different from π ππ π₯ = 2 ? CL 6-163. Solve each of the following equations on the domain [0, 2π]. a. 2 πππ π₯ + 1 = 0 b. π ππ 2 3 π₯=4 c. π‘ππ π₯ = β1 CL 6-164. Find a function in terms of sine and another in terms of cosine that will generate the graphs below. a. b. CL 6-165. Given ΞABC with β A = 34°, = 8cm, and = 6cm. a. Draw two different possible triangles that would have the measurements described for ΞABC. Why are there two possible solutions? b. Find two possible lengths for side . c. What is the difference in the area of the two triangles? CL 6-166. Find the angle of inclination for each line below. 2 a. π¦ = 3 π₯ β 1 b. π¦ = β2π₯ + 5 c. What is the measure of the acute angle that is formed at the intersection of the two lines? CL 6-167. π¦ = π ππ β1 π₯, π¦ = πππ β1 π₯, and π¦ have restricted ranges. Give the range of each of them. CL 6-168. Given: π ππ π΄ 5 = π‘ππ 5 β1 π₯ are all functions because they π = 13and π ππ π΅ = 4, where 0 < π΄, π΅ < 2 . a. What is the exact value of cos A? b. What is the exact value of sinB? c. Find the exact value of cos(A + B). CL 6-169. Simplify each of the following trig expressions. a. 20 π ππ(2π₯) πππ (2π₯) b. π ππ(π β π₯) c. π ππ 2 π β πππ 2 π π d. πππ (π₯ + 2 ) CL 6-170. Solve the following equations in the domain stated. a. 2 πππ 2 π₯ + π ππ π₯ = 2, [0, 2π] b. 2 πππ 2 π₯ + π ππ π₯ = 2 c. π ππ π₯ β π ππ(2π₯) = 0, [0, 2π] d. π ππ π₯ β π ππ(2π₯) = 0 CL 6-171. A spring is hanging from the ceiling of a room. It is pulled and released so that the distance from the floor with respect to time is a sinusoidal motion. At t = 2.5 seconds, the spring is at a minimum from the floor, 1.5 feet. At t = 3.75 seconds, the spring is at a maximum, 6 feet. Find a trigonometric equation that models the motion of the spring as a function of time. a. Find the height of the spring from the floor at t = 3 seconds. b. Find the first two times when the spring is 2 feet from the floor. CL 6-172. Solve each of the following equations. a. b. c. d. 18(1.03) π₯ β 20 = 300 πππ2 5 + πππ2 π₯ = 3 8π₯ 2.7 = 160 πππ3 (π₯ + 5) β πππ3 (π₯ β 1) = 1 CL 6-173. Find the equation of the line π¦ = ππ₯ + π so that π(π₯) is continuous. CL 6-174. Check your answers using the following table. Which problems do you feel confident about? Which problems were hard? Have you worked on problems like these in math classes you have taken before? Use the table to make a list of topics with which you need help or more practice. Answers and Support for Closure Problems What Have I Learned? Problem Solution Need Help? CL 6-162. Lesson 6.1.1 Math Note Problems 6-1, 6-2, 6-9, 6-24, and 6-25 a. , the function sinβ1 x can only have one output. c. CL 6-163. CL 6-164. b. a. , b. , c. a. y = 2sin(2x) + 2, y = 2cos(2(x β β3sin( (x β )) β1, y = 3cos( (x + CL 6-165. )) + 2, Lessons 6.1.1 and 6.1.4 Problems 6-5, 6-10, 6-26,6-27, 6-48, 6-56, and 6-99 , b. y = Lesson 6.2.1 Math Note Problems 6-65 to 6-69, 6-72, 6-79, and 6-85 )) β 1 Lesson 1.3.1 Lesson 6.1.3 Math Note a. b. 10.63 cm and 2.63 cm c. 17.89 cm2 CL 6-166. CL 6-167. CL 6-168. a. 33.7° b. β63.4° c. 82.9° sinβ1: a. b. c. Problems 6-38, 6-40, and6113 Lesson 6.1.4 Math Note Problems 6-52, 6-53, and6-55 , cosβ1: [0, Ο], tanβ1: Lessons 6.1.2 and 6.1.4 Problems 6-20, 6-22, 6-23, 629, 6-50, and 6-58 Lesson 4.1.2 6.2.2 Math Note Problems 6-81, 6-82, 6-83, 6-96, 6-98, 6-109, and 6-138 CL 6-169. a. b. c. d. 10 sin 4x sin x βcos(2ΞΈ) βsin x CL 6-170. a. 0, Ο, 2Ο, b. 0, Ο, c. CL 6-171. , Lesson 6.3.2 Math Note Problems 6-117, 6-119, and 6-129 , , 0, Ο, 2Ο, d. 0, Ο, Lesson 6.2.2 Math Note Lesson 6.3.1 Math Note Problems 6-83, 6-87, 6-88, 6-95, 6-106, 6-111, and 6-114 , all + 2Οn , , all + 2Οn y = 2.25 sin(0.8Ο(x β 3.125)) + 3.75 a. 3.055 feet b. At 0.2704 and 2.2296 seconds CL 6-172. a. b. c. d. 97.364 8/5 3.033 4 CL 6-173. y = βx β 6 Lesson 6.3.2 Problems 6-11, 6-71, 6-84, 6-85, 6-92, and 6-93 Lesson 3.3.2 Math Note Problems 3-122 to 3-126, and 6-103 Lesson 5.2.4 Math Note Problem 5-93 and 5-100