13-1 Trigonometric Identities. Find the exact value of each expression if 0 < θ < If cot θ = 2, find tan θ. SOLUTION: 2. If, find cos θ.
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1 Find the exact value of each expression if 0 < θ < If cot θ = 2, find tan θ. 2. If, find cos θ. Since is in the first quadrant, is positive. Thus,. 3. If, find sin θ. Since is in the first quadrant, is positive. Thus,. esolutions Manual - Powered by Cognero Page 1
2 4. If, find csc θ. Since is in the first quadrant, is positive. Thus,. Rationalize the denominator. Simplify each expression 5. tan θ cos 2 θ 6. csc 2 θ cot 2 θ esolutions Manual - Powered by Cognero Page 2
3 7. esolutions Manual - Powered by Cognero Page 3
4 8. CCSS PERSEVERANCE When unpolarized light passes through polarized sunglass lenses, the intensity of the light is cut in half. If the light then passes through another polarized lens with its axis at an angle of θ to the first, the intensity of the emerging light can be found by using the formula, where I o is the intensity of the light incoming to the second polarized lens, I is the intensity of the emerging light, and θ is the angle between the axes of polarization. a. Simplify the formula in terms of cos θ. b. Use the simplified formula to determine the intensity of light that passes through a second polarizing lens with axis at 30 to the original. a. b.substitute 30 for θ. The light has three-fourths the intensity it had before passing through the second polarizing lens. esolutions Manual - Powered by Cognero Page 4
5 Find the exact value of each expression 0 < θ < If, find csc θ. Since is in the first quadrant, is positive. Thus,. esolutions Manual - Powered by Cognero Page 5
6 10. If, find tan θ. Since is in the first quadrant, is positive. Thus,. 11. If, find cos θ. Since is in the first quadrant, is positive. Thus,. esolutions Manual - Powered by Cognero Page 6
7 12. If tan θ = 2, find sec θ. Since is in the first quadrant, is positive. Thus,. Find the exact value of each expression 180 < θ < If, find csc θ. Since θ is in the third quadrant, is negative. Therefore,. 14. If, find tan θ. Since θ is in the third quadrant, Therefore,. is positive. esolutions Manual - Powered by Cognero Page 7
8 15. If, find csc θ. Since θ is in the third quadrant, is negative. Therefore,. 16. If, find cos θ. Since θ is in the third quadrant, is negative. Therefore,. esolutions Manual - Powered by Cognero Page 8
9 Find the exact value of each expression 270 < θ < If, find sin θ. Since θ is in the fourth quadrant, is negative. Therefore,. 18. If, find sec θ. Since θ is in the fourth quadrant, Therefore,. is positive. 19. If, find cos θ. esolutions Manual - Powered by Cognero Page 9
10 20. If, find cos θ. Since θ is in the fourth quadrant, is positive. Therefore,. Simplify each expression esolutions Manual - Powered by Cognero Page 10
11 ELECTRONICS When there is a current in a wire in a magnetic field, such as in a hairdryer, a force acts on the wire. The strength of the magnetic field can be determined using the formula, where F is the force on the wire, I is the current in the wire, l is the length of the wire, and θ is the angle the wire makes with the magnetic field. Rewrite the equation in terms of sin θ (Hint : Solve for F.) esolutions Manual - Powered by Cognero Page 11
12 28. Simplify each expression esolutions Manual - Powered by Cognero Page 12
13 34. SUN The ability of an object to absorb energy is related to a factor called the emissivity e of the object. The emissivity can be calculated by using the formula, where W is the rate at which a person s skin absorbs energy from the Sun, S is the energy from the Sun in watts per square meter, A is the surface area exposed to the Sun, and θ is the angle between the Sun s rays and a line perpendicular to the body. a. Solve the equation for W. Write your answer using only sin θ or cos θ. b. Find W if e = 0.80, θ = 40, A = 0.75m 2, and S = 1000 W/m2. Round to the nearest hundredth. a. b. Substitute the values of e, θ, A and S and evaluate. 35. CCSS MODELING The map shows some of the buildings in Maria s neighborhood that she visits on a regular basis. The sine of the angle θ formed by the roads connecting the dance studio, the school, and Maria s house is. a. What is the cosine of the angle? b. What is the tangent of the angle? c. What are the sine, cosine, and tangent of the angle formed by the roads connecting the piano teacher s house, the school, and Maria s house? a. Given. esolutions Manual - Powered by Cognero Page 13
14 b. Given. c. 36. MULTIPLE REPRESENTATIONS In this problem, you will use a graphing calculator to determine whether an equation may be a trigonometric identity. Consider the trigonometric identity tan 2 θ sin 2 θ = tan 2 θ sin 2 θ. a. TABULAR Complete the table. esolutions Manual - Powered by Cognero Page 14
15 b. GRAPHICAL Use a graphing calculator to graph tan 2 θ sin 2 θ = tan 2 θ sin 2 θ as two separate functions. Sketch the graph. c. ANALYTICAL If the graphs of the two functions do not match, then the equation is not an identity. Do the graphs coincide? d. ANALYTICAL Use a graphing calculator to determine whether the equation sec 2 x 1 = sin 2 x sec 2 x may be an identity. (Be sure your calculator is in degree mode.) a. b. KEYSTROKES: Y= TAN ALPHA [x] ) x 2 SIN ALPHA [x] ) x 2 ENTER TAN ALPHA [x] ) x 2 SIN ALPHA [x] ) x 2 GRAPH c. yes esolutions Manual - Powered by Cognero Page 15
16 d. Plug in sec 2 x 1 for Y1 and sin 2 x sec 2 x for Y2 in a graphing calculator and form a table. From the table, the values of sec 2 x 1 and sin 2 x sec 2 x are the same. Therefore, the equation is an identity. 37. SKIING A skier of mass m descends a θ-degree hill at a constant speed. When Newton s laws are applied to the situation, the following system of equations is produced: and, where g is the acceleration due to gravity, is the normal force exerted on the skier, and is the coefficient of friction. Use the system to define as a function of θ. Substitute for F n and solve for. esolutions Manual - Powered by Cognero Page 16
17 Simplify each Expression esolutions Manual - Powered by Cognero Page 17
18 42. CCSS CRITIQUE Clyde and Rosalina are debating whether an equation from their homework assignment is an identity. Clyde says that since he has tried ten specific values for the variable and all of them worked, it must be an identity. Rosalina argues that specific values could only be used as counterexamples to prove that an equation is not an identity. Is either of them correct? Explain your reasoning. Rosalina; there may be other values for which the equation is not true. 43. CHALLENGE Find a counterexample to show that is not an identity. Sample answer: x = REASONING Demonstrate how the formula about illuminance from the beginning of the lesson can be rewritten to show that. 45. WRITING IN MATH Pythagoras is most famous for the Pythagorean Theorem. The identity is an example of a Pythagorean identity. Why do you think that this identity is classified in this way? Sample answer: The functions and can be though of as the lengths of the legs of a right triangle, and the number 1 can be thought of as the measure of the corresponding hypotenuse. 46. PROOF Prove that by using the quotient and negative angle identities. 47. OPEN ENDED Write two expressions that are equivalent to. Sample answer: and esolutions Manual - Powered by Cognero Page 18
19 48. REASONING Explain how you can use division to rewrite as. Divide all of the terms by. 49. CHALLENGE Find if and. Since θ is in the second quadrant, cot θ is negative. Therefore,. 50. ERROR ANALYSIS Jordan and Ebony are simplifying. Is either of them correct? Explain your reasoning. Ebony; Jordan did not use the identity that and made an error adding rational expressions. esolutions Manual - Powered by Cognero Page 19
20 51. Refer to the figure below. If what is the length of? A 5 B 4 C 3.2 D Option A is the correct answer. 52. PROBABILITY There are 16 green marbles, 2 red marbles, and 6 yellow marbles in a jar. How many yellow marbles need to be added to the jar in order to double the probability of selecting a yellow marble? F 4 G 6 H 8 J 12 The probability of getting a yellow marble is. To double the probability of selecting a yellow marble, we need to add x marbles. That is: Option J is the correct answer. esolutions Manual - Powered by Cognero Page 20
21 53. SAT/ACT Ella is 6 years younger than Amanda. Zoe is twice as old as Amanda. The total of their ages is 54. Which equation can be used to find Amanda s age? A B C D E Let x be the age of Amanda. Therefore, the ages of Ella, Amanda and Zoe are x 6, x, 2x. Given:. Option D is the correct answer. 54. Which of the following functions represents exponential growth? F G H. J To be an exponential growth, the value in the parenthesis must be greater than one. The variable will be in the exponent. Therefore, option G is the correct answer. Find each value. Write angle measures in radians. Round to the nearest hundredth. 55. Use a calculator. KEYSTROKES: 2nd [COS 1 ] ( ) 1 2 ) ENTER. 56. Use a calculator. KEYSTROKES: 2nd [SIN 1 ] 2nd [π] 2 ) ENTER. esolutions Manual - Powered by Cognero Page 21
22 57. Use a calculator. KEYSTROKES: 2nd [TAN 1 ] 2nd [ ] 3 ) 3 ) ENTER. 58. Use a calculator. KEYSTROKES: TAN 2nd [COS 1 ] 6 7 ) ENTER. 59. Use a calculator. KEYSTROKES: SIN 2nd [TAN 1 ] 2nd [ ] 3 ) 3 ) ENTER. 60. Use a calculator. KEYSTROKES: COS 2nd [SIN 1 ] 3 5 ) ENTER. esolutions Manual - Powered by Cognero Page 22
23 61. PHYSICS The weight is attached to a spring and suspended from the ceiling. At equilibrium, the weight is located 4 feet above the floor. The weight is pulled down 1 foot and released. Write the equation for the distance d of the weight above the floor as a function of the time t seconds assuming that the weight returns to its lowest position every 4 seconds. An equation for the function is. At equilibrium, the weight is 4 inches above the floor. Therefore, the vertical shift is k = 4. The weight is 1 foot closer to the floor at its lowest point, so the amplitude a is 1. The weight returns to its lowest position every 4 seconds, therefore the period is 4. There is no horizontal shift. So, or. 62. Evaluate the sum of each geometric series. There are or 5 terms. Find the sum. esolutions Manual - Powered by Cognero Page 23
24 63. There are or 7 terms. Find the sum. 64. There are or 8 terms. Find the sum. esolutions Manual - Powered by Cognero Page 24
25 65. Solve each equation esolutions Manual - Powered by Cognero Page 25
13-1 Trigonometric Identities. Find the exact value of each expression if 0 < θ < If cot θ = 2, find tan θ. ANSWER: 2. If, find cos θ.
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