Hmwk 14 over sec. 3.3

1/1/24, 5:32 PM Hmwk 14 over sec. 3.3-Rachel Chrisman https://tdx.acs.pearsonprd.tech/api/v1/print/highered 1/27 1. 2. 3. Student: Rachel Chrisman Date: 01/01/24 Instructor: Robin Ayers Course: MATH 117 Trigonometry On Demand July 2023 - July 2025 Assignment: Hmwk 14 over sec. 3.3 Which of the following is not a valid double-angle formula for cosine? Choose the correct answer. A. cos 2 = 2 cos − 1 2 B. cos 2 = 1 − 2 sin 2 C. cos 2 = cos − sin 2 2 D. cos 2 = 2 sin − 1 2 Which of the following is not a valid power reduction formula? Choose the correct answer. A. tan = 2 1 − cos 2 1 + cos 2 B. csc = 2 1 − sin 2 2 C. cos = 2 1 + cos 2 2 D. sin = 2 1 − cos 2 2 If is an angle with a terminal side lying in Quadrant III, then which of the following expressions is equivalent to ? 2 sin 2 Choose the correct answer. A. − 1 + cos 2 B. 1 + cos 2 C. 1 − cos 2 D. − 1 − cos 2

1/1/24, 5:32 PM Hmwk 14 over sec. 3.3-Rachel Chrisman https://tdx.acs.pearsonprd.tech/api/v1/print/highered 2/27 4. 5. 6. 7. If is an angle with a terminal side lying in Quadrant IV, then which of the following expressions is not equivalent to ? 2 tan 2 Choose the correct answer. A. 1 − cos sin B. 1 − cos 1 + cos C. sin 1 + cos D. − 1 − cos 1 + cos Which of the following can best be evaluated using the half-angle formula for the sine function? Choose the correct answer. A. sin tan − 1 1 + cos − 1 3 2 B. sin cos − 1 − 1 2 C. sin 1 2 cos − 1 ( ) − 2 D. sin 1 2 cos − 1 − 2 3 Simplify the expression. − 1 − 2 10 10 − 1 − 2 10 10 = (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) Simplify the expression. − 1 − 11 23 1 + 11 23 − 1 − 11 23 1 + 11 23 = (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)

1/1/24, 5:32 PM Hmwk 14 over sec. 3.3-Rachel Chrisman https://tdx.acs.pearsonprd.tech/api/v1/print/highered 4/27 11. 12. 13. Rewrite the expression as the sine, cosine, or tangent of a double-angle. Then find the exact value of the trigonometric expression without the use of a calculator. 2 sin ( ) cos ( ) − 382.5 ° − 382.5 ° Use the appropriate double-angle formula to rewrite the given expression as the sine, cosine, or tangent of the corresponding double-angle. 2 sin ( ) cos ( ) − 382.5 ° − 382.5 ° = (Do not evaluate. Type your answer in degrees. Do not include the degree symbol in your answer.) Find the exact value of the trigonometric expression without the use of a calculator. 2 sin ( ) cos ( ) − 382.5 ° − 382.5 ° = (Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.) Rewrite the expression as the sine, cosine, or tangent of a double-angle. Then find the exact value of the trigonometric expression without the use of a calculator. 1 − 2 sin 2 − 11 12 Use the appropriate double-angle formula to rewrite the given expression as the sine, cosine, or tangent of the corresponding double-angle. 1 − 2 sin 2 − 11 12 = (Do not evaluate. Use integers or fractions for any numbers in the expression. Type your answer in radians.) Find the exact value of the trigonometric expression without the use of a calculator. 1 − 2 sin 2 − 11 12 = (Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.) Rewrite the expression as the sine, cosine, or tangent of a double-angle. Then find the exact value of the trigonometric expression without the use of a calculator. 2 tan (60 ° ) 1 − tan 2 (60 ° ) Use the appropriate double-angle formula to rewrite the given expression as the sine, cosine, or tangent of the corresponding double-angle. Choose the appropriate double-angle formula below. 2 tan (60 ° ) 1 − tan 2 (60 ° ) = (Do not evaluate. Type your answer in degrees. Do not include the degree symbol in your answer.) Find the exact value of the trigonometric expression without the use of a calculator. 2 tan (60 ° ) 1 − tan 2 (60 ° ) = (Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.)

1/1/24, 5:32 PM Hmwk 14 over sec. 3.3-Rachel Chrisman https://tdx.acs.pearsonprd.tech/api/v1/print/highered 5/27 14. 15. Rewrite the expression as the sine, cosine, or tangent of a double-angle. Then find the exact value of the trigonometric expression without the use of a calculator. 1 − 2 sin (165 ° ) 2 Use the appropriate double-angle formula to rewrite the given expression as the sine, cosine, or tangent of the corresponding double-angle. 1 − 2 sin (165 ° ) 2 = (Do not evaluate. Type your answer in degrees. Do not include the degree symbol in your answer.) Find the exact value of the trigonometric expression without the use of a calculator. 1 − 2 sin (165 ° ) 2 = (Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.) Rewrite the expression as the sine, cosine, or tangent of a double-angle. Then find the exact value of the trigonometric expression without the use of a calculator. 2 cos − 1 2 13 12 Use the appropriate double-angle formula to rewrite the given expression as the sine, cosine, or tangent of the corresponding double-angle. 2 cos − 1 2 13 12 = (Do not evaluate. Use integers or fractions for any numbers in the expression. Type your answer in radians.) Find the exact value of the trigonometric expression without the use of a calculator. 2 cos − 1 2 13 12 = (Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.)

1/1/24, 5:32 PM Hmwk 14 over sec. 3.3-Rachel Chrisman https://tdx.acs.pearsonprd.tech/api/v1/print/highered 7/27 17. 18. Suppose that the terminal side of angle lies in Quadrant III such that . Determine the value of . sin = − 8 17 sin 2 One or more of the equations below represent valid double-angle formulas for . sin 2 sin 2 = cos − sin 2 2 sin 2 = 2 tan 1 − tan 2 sin 2 = tan 1 + tan 2 sin 2 = cos + sin 2 2 sin 2 = 2 cos − 1 2 sin 2 = 2 sin cos Drag all equations that represent a valid formula for into the box provided. sin 2 Choose and complete the correct formula below by typing the correct value for each trigonometric expression. Enter these values in the order seen in the previous step. Select the correct choice below and fill in the answer boxes within your choice. (Simplify your answers, including any radicals. Use integers or fractions for any numbers in the expression.) A. sin 2 = 2 1 − 2 B. sin 2 = 2 C. sin 2 = + 2 2 D. sin 2 = 1 + 2 E. sin 2 = 2 − 1 2 F. sin 2 = − 2 2 Type the exact value of . sin 2 = sin 2 (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) Use the given information to determine the values of , , and . sin cos tan ; sin 2 = 19 181 < 2 < 2 sin = (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) cos = (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) tan = (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)

1/1/24, 5:32 PM Hmwk 14 over sec. 3.3-Rachel Chrisman https://tdx.acs.pearsonprd.tech/api/v1/print/highered 8/27 19. 20. 21. 22. Use the given information to determine the values of , , and . sin cos tan ; tan 2 = 11 3 < 2 < 3 2 sin = (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) cos = (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) tan = (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) Rewrite the given function as an equivalent function containing only cosine terms raised to a power of 1. f(x) = − 9 sin x 2 f(x) = (Type an exact answer in simplified form. Use integers or fractions for any numbers in the expression. Do not factor.) Use the power-reducing formulas to rewrite the expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1. f(x) = 5 sin x 4 f(x) = (Type an exact answer in simplified form. Use integers or fractions for any numbers in the expression. Do not factor.) Rewrite the given function as an equivalent function containing only cosine terms raised to a power of 1. f(x) = − 4 sin (x) cos (x) 2 2 f(x) = (Type an exact answer in simplified form. Use integers or fractions for any numbers in the expression. Do not factor.)

1/1/24, 5:32 PM Hmwk 14 over sec. 3.3-Rachel Chrisman https://tdx.acs.pearsonprd.tech/api/v1/print/highered 10/27 24. (1) II. I. III. IV. (2) positive negative (3) Use a half-angle formula to evaluate the expression without using a calculator. sin ( − 112.5 ° ) a. What is the measure of the half -angle ? 2 2 = ° (Type an integer or a decimal. Do not include the degree symbol in your answer.) b. What is the measure of angle ? = ° (Type an integer or a decimal. Do not include the degree symbol in your answer.) c. The terminal side of the half -angle lies in Quadrant (1) 2 d. Is the sine function positive or negative in this quadrant? The sine function is (2) in this quadrant. e. Which half -angle formula should be used to evaluate the expression ? sin ( − 112.5 ° ) The half -angle formula (3) should be used to evaluate the expression . sin ( − 112.5 ° ) f. Choose the correct formula below used to evaluate the expression and complete the expression in the numerator of the expression under the radical. sin ( − 112.5 ° ) (Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) A. sin ( − 112.5 ° ) = 2 B. sin ( − 112.5 ° ) = − 2 sin = − 2 1 − cos ( ) 2 sin = 2 1 − cos ( ) 2

1/1/24, 5:32 PM Hmwk 14 over sec. 3.3-Rachel Chrisman https://tdx.acs.pearsonprd.tech/api/v1/print/highered 11/27 25. (1) III. I. IV. II. (2) negative positive (3) Use a half-angle formula to evaluate the expression without using a calculator. cos ( − 75 ° ) a. What is the measure of the half -angle ? 2 2 = ° (Use integers or decimals for any numbers in the expression.) b. What is the measure of angle ? = ° (Use integers or decimals for any numbers in the expression.) c. The terminal side of the half -angle lies in Quadrant (1) 2 d. Is the cosine function positive or negative in this quadrant? The cosine function is (2) in this quadrant. e. Which half -angle formula should be used to evaluate the expression ? cos ( − 75 ° ) The half -angle formula (3) should be used to evaluate the expression . cos ( − 75 ° ) f. Choose the correct formula below used to evaluate the expression and complete the expression in the numerator of the expression under the radical. cos ( − 75 ° ) (Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) A. cos ( − 75 ° ) = − 2 B. cos ( − 75 ° ) = 2 g. Completely simplify the expression from part f . cos ( − 75 ° ) = (Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) cos = 2 1 + cos ( ) 2 cos = − 2 1 + cos ( ) 2

1/1/24, 5:32 PM Hmwk 14 over sec. 3.3-Rachel Chrisman https://tdx.acs.pearsonprd.tech/api/v1/print/highered 13/27 27. (1) I. II. III. IV. (2) positive negative (3) Use a half-angle formula to evaluate the expression without using a calculator. sec − 11 8 a. What is the measure of the half -angle ? 2 2 = (Type an exact answer in terms of . Use integers or fractions for any numbers in the expression.) b. What is the measure of angle ? = (Type an exact answer in terms of . Use integers or fractions for any numbers in the expression.) c. The terminal side of the half -angle lies in Quadrant (1) 2 d. Is the secant function (and cosine function) positive or negative in this quadrant? The secant function (and cosine function) are (2) in this quadrant. e. Which half -angle formula should be used to evaluate the expression ? sec − 11 8 The half-angle formula (3) should be used to evaluate the expression . sec − 11 8 f. Choose the correct formula below used to evaluate the expression and complete the expression in the numerator of the expression under the radical. sec − 11 8 (Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) A. sec − 11 8 = − 1 2 B. sec − 11 8 = 1 2 g. Completely simplify the expression from part f . sec − 11 8 = (Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) sec = = − 2 1 cos 2 1 1 + cos 2 sec = = 2 1 cos 2 1 1 + cos 2

1/1/24, 5:32 PM Hmwk 14 over sec. 3.3-Rachel Chrisman https://tdx.acs.pearsonprd.tech/api/v1/print/highered 14/27 28. (1) Quadrant IV. Quadrant I. Quadrant II. Quadrant III. (2) positive. negative. (3) Suppose that such that . Determine the value of . cos = − 8 17 < < 2 sin 2 In which quadrant does the terminal side of angle lie? 2 The terminal side of angle lies in (1) 2 Is the value of the sine of an angle positive or negative in the correct quadrant from the previous step? The sine of an angle is (2) Choose the correct formula for based on the given information. sin 2 (3) sin 2 = Choose and complete the correct formula below by typing the correct value in each set of parentheses. Each pair of parentheses represents the exact value of seen in the correct half-angle formula from the previous step. Select the correct choice below and fill in the answer box(es) to complete your choice. cos (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) A. sin = − 2 1 − 2 B. sin = − 2 1 − 1 + C. sin = 2 1 − 1 + D. sin = 2 1 − 2 E. sin = − 2 1 + 2 F. sin = 2 1 + 2 1 − cos 1 + cos − 1 + cos 2 − 1 − cos 2 1 − cos 2 − 1 − cos 1 + cos 1 + cos 2

1/1/24, 5:32 PM Hmwk 14 over sec. 3.3-Rachel Chrisman https://tdx.acs.pearsonprd.tech/api/v1/print/highered 16/27 30. (1) (2) (3) Suppose that such that . Determine the value of . cos = 11 13 < < 2 3 2 sin 2 In which quadrant does the terminal side of angle lie? 2 The terminal side of angle lies in (1) 2 Is the value of the sine of an angle positive or negative in the correct quadrant from the previous step? The sine of an angle is (2) Choose the correct formula for based on the given information. sin 2 (3) sin 2 = Choose and complete the correct formula below by typing the correct value in each set of parentheses. Each pair of parentheses represents the exact value of seen in the correct half-angle formula from the previous step. Select the correct choice below and fill in the answer box(es) to complete your choice.(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) cos A. sin 2 = − 1 − 1 + B. sin 2 = − 1 − 2 C. sin 2 = 1 + 2 D. sin 2 = 1 − 1 + E. sin 2 = 1 − 2 F. sin 2 = − 1 + 2 Quadrant I. Quadrant III. Quadrant II. Quadrant IV. positive. negative. − 1 − cos 2 1 − cos 2 1 + cos 2 − 1 − cos 1 + cos − 1 + cos 2 1 − cos 1 + cos

1/1/24, 5:32 PM Hmwk 14 over sec. 3.3-Rachel Chrisman https://tdx.acs.pearsonprd.tech/api/v1/print/highered 17/27 31. 32. 33. Use a half-angle formula to evaluate the expression without using a calculator. sin 5 12 Choose the correct formula below used to evaluate the expression and complete the expression in the numerator of the expression under the radical. sin 5 12 (Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) A. sin 5 12 = 2 B. sin 5 12 = − 2 Use a half-angle formula to evaluate the expression without a calculator. tan (157.5 ° ) Choose the one formula below that can be used to evaluate the expression and complete the expression. tan (157.5 ° ) (Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) A. tan (157.5 ° ) = − 1 − 2 B. tan (157.5 ° ) = 1 − 2 C. tan (157.5 ° ) = 1 − 1 + D. tan (157.5 ° ) = 1 + Use a half-angle formula to evaluate the expression without using a calculator. cos 3 8 Choose the correct formula below used to evaluate the expression and complete the expression in the numerator of the expression under the radical. cos 3 8 (Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) A. cos 3 8 = 2 B. cos 3 8 = − 2

1/1/24, 5:32 PM Hmwk 14 over sec. 3.3-Rachel Chrisman https://tdx.acs.pearsonprd.tech/api/v1/print/highered 19/27 39. 40. 41. Verify the identity. − 2 cot 1 + cot 2 = − sin 2 Which of the following four statements establishes the identity? A. − 2 cot 1 + cot 2 = − 2 cot csc 2 = sin − cos 2 2 = − sin 2 B. − 2 cot 1 + cot 2 = − 2 cot sec 2 = − 2 cos sin = − sin 2 C. − 2 cot 1 + cot 2 = − 2 cot sec 2 = sin − cos 2 2 = − sin 2 D. − 2 cot 1 + cot 2 = − 2 cot csc 2 = − 2 cos sin = − sin 2 Verify the identity. tan = sin 2 1 + cos 2 Which of the following three statements verifies the given equation? A. = = = tan sin 2 1 + cos 2 2 sin cos 1 + 2 sin 2 − 1 2 sin cos 2 sin 2 B. = = = tan sin 2 1 + cos 2 2 sin cos 1 + 1 + 2 cos 2 2 sin cos 2 cos 2 C. = = = tan sin 2 1 + cos 2 2 sin cos 1 + 2 cos 2 − 1 2 sin cos 2 cos 2 Verify the following identity. tan = 2 sec − 1 tan Which of the following statements establishes the identity? A. tan 2 = 1 + cos 2 = sec − 1 tan B. tan 2 = 1 − cos 2 = sec − 1 tan C. tan 2 = 1 − cos sin = sec − 1 tan

1/1/24, 5:32 PM Hmwk 14 over sec. 3.3-Rachel Chrisman https://tdx.acs.pearsonprd.tech/api/v1/print/highered 20/27 42. 43. 44. Verify the identity. cot − cot 2 = csc Which of the following four statements establishes the identity? A. cot − cot 2 = − 1 + cos sin cos sin = csc B. cot − cot 2 = − 1 − cos sin sin cos = csc C. cot − cot 2 = − 1 − cos sin cos sin = csc D. cot − cot 2 = − 1 + cos sin sin cos = csc Use a double-angle formula to find the exact value of the following expression without using a calculator. cos 2 sin − 1 1 2 cos 2 sin − 1 1 2 = (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) Use a double-angle formula to find the exact value of the following expression without using a calculator. cos 2 tan − 1 2 5 cos 2 tan − 1 2 5 = (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)