Principles of Math 12 - Trigonometry II Practice Exam

Principles of Math 12 - Trigonometry II Practice Exam -...

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Unformatted text preview: Principles of Math 12 - Trigonometry II Practice Exam 1 www.math12.com Trigonometry II Practice Exam Use this sheet to record your answers 1. NR 2. 19. 28. 2. NR 3. 20. NR 7. NR 1. 11. 21. 29. 3. 12. NR 5. 30. 4. 13. 22. 31. 5. 14. 23. 32. 6. NR 4. 24. 33. 7. 15. 25. 8. 16. NR 6. 9. 17. 26. 10. 18. 27. Copyright of Math 12y Trigonometry II Practice Exam Principles © Barr - Mabillard, 2006 2 www.math12.com www.math12.com Trigonometry II Practice Exam 1. The exact value of sin 75˚ can be determined using the expression A. B. C. D. sin 90˚ - sin 15˚ sin 45˚ + sin 30˚ sin 45˚cos 30˚ + cos 45˚sin 30˚ cos 45˚cos 30˚ + cos 45˚cos 30˚ Use the following information to answer the next question. A student solves the equation tan x = −1 in their graphing calculator as shown in the diagram below. The student determines the general solution of this graph is − 2. π 4 + nπ , n ∈ I The general solution to the equation tan ( 5 x ) = −1 is A. − π 3 π + nπ , n∈ I 10 + nπ , n ∈ I 4 5π C. − + 5nπ , n ∈ I 4 π nπ D. − + , n∈ I 20 5 B. − Principles of Math 12 - Trigonometry II Practice Exam 3 www.math12.com Numerical Response 1. cos 2 x + sin x may be verified by substituting 2.1 rad sin 2 x for x on each side. When this substitution is made, the numerical value of each side, to the nearest hundredth, is _________. The identity cot 2 x + csc x = Use the following information to answer the next question. A Ferris wheel at an amusement park, with a diameter of 12 m, can be π t + 9 , where h ( t ) is the 20 height above the ground in metres, and t is the time in seconds. modeled using the equation h(t ) = −6 cos 3. The number of seconds required for a rider to reach a height of 14 m for the first time is, to the nearest tenth, A. B. C. D. 4. 16.3 s 16.5 s 20.4 s 932 s The expression cot 2 x + csc x − 4 is equivalent to A. csc 2 x + csc x − 5 B. csc 2 x + csc x − 3 cos 2 x 1 C. + −4 2 sin x cos x D. -4 Principles of Math 12 - Trigonometry II Practice Exam 4 www.math12.com Use the following information to answer the next question. The graph of f ( x ) = cos 2 x + cos x + 1 is shown below 5. If the equation cos 2 x + cos x + 1 = 3 has the general solution 2nπ , n ∈ I , ⎛ x⎞ ⎛ x⎞ then a possible solution to the equation cos 2 ⎜ ⎟ + cos ⎜ ⎟ + 1 = 3 is ⎝3⎠ ⎝3⎠ A. 2π B. 3π C. 9π D. 12π 6. If sin A = A. B. C. D. m m2 and tan A = 3 , where m, n ≠ 0, then cos A is equivalent to n n n2 m m3 n4 mn 2 1 mn 2 Principles of Math 12 - Trigonometry II Practice Exam 5 www.math12.com 7. The expression A. 1 + tan 2 x , is equivalent to 1 − sin 2 x (1 + tan x )(1 − tan x ) (1 + sin x )(1 − sin x ) B. 1 C. sec x D. sec 2 x 8. If tan 2 x = 5 , then sec 2 x is equivalent to 7 12 7 7 B. 5 5 74 C. 74 74 D. 7 A. 9. The expression cos 2 ( 4π ) − sin 2 ( 4π ) is equivalent to A. cos 2 ( 4π ) B. sin 2 ( 8π ) C. cos ( 8π ) D. cos ( 4π ) sin ( 4π ) Principles of Math 12 - Trigonometry II Practice Exam 6 www.math12.com Use the following information to answer the next question. The graphs of f ( x ) = cos x and g ( x ) = cos ( 2 x ) are shown below The graphs intersect four times on the interval 0 ≤ x ≤ 2π 10. If the domain is changed to 0 < x < 2π , (the equality has been removed) a correct statement is A. B. C. D. There are more solutions There are fewer solutions There are the same number of solutions There is no change in the number of solutions Numerical Response 2. The equation csc 2 x − 2 = cos 2 x has four solutions in the interval 0 < x < 2π . The number of solutions for x in the interval 0 < x < 14π is _________. Principles of Math 12 - Trigonometry II Practice Exam 7 www.math12.com Use the following information to answer the next question. The steps used by a student to simplify the expression ( sin x + cos x ) are 2 shown below Step 1 : sin 2 x + cos 2 x Step 2 : sin 2 x + (1 − sin 2 x ) Step 3 : (1- cos x ) + (1 − sin x ) Step 4 : 2 - sin 2 x − cos 2 x 2 2 Numerical Response 3. 11. The step which contains a mathematical error is step _______. The value of m in the equation m sin x cot x = 8 is 4 csc x tan x sin x cos x 32 2 sin x cos x B. tan x C. 32 sec 2 x D. 32sec x csc x A. 12. The solutions to the equation cos 2 x = cos x , where 0 ≤ x < 2π are A. 0, B. π 2 ,π , π 3π 3π , 2π 2 , 22 C. 0, π 3π 2 , 2 π 3π D. 0, , , 2π 22 Principles of Math 12 - Trigonometry II Practice Exam 8 www.math12.com 13. The expression A. B. C. π 6 π 6 π ± 2nπ , 5π ± 2nπ 6 ± nπ ± 2nπ , 6 nπ D. 2 14. cos x is undefined when the values of x are 1 − 2sin x 5π π ± 2nπ , ± nπ 6 2 π⎞ ⎛ The expression sec ⎜ x − ⎟ is equivalent to 2⎠ ⎝ π A. sec x − sec 2 π⎞ ⎛ B. cos ⎜ x − ⎟ 2⎠ ⎝ C. csc x ⎛π ⎞ D. − sin ⎜ − x ⎟ ⎝2 ⎠ Numerical Response 4. 15. 1 π = 0.43 , and 0 ≤ x < , then the value of x in radians, to the 2 2 1 + cot x nearest tenth, is ________. If The expression sin x 1 is equivalent to + tan x sec x A. 2 cos x B. 2 sec x sin x + 1 C. tan x + sec x sin x D. tan x sec x Principles of Math 12 - Trigonometry II Practice Exam 9 www.math12.com 16. 7 4 and cos B = , where A and B are acute angles, the value of 8 5 cos ( A − B ) is equal to Given sin A = 3 8 16 49 B. + 25 64 4 15 + 21 C. 40 28 − 3 15 D. 40 A. 17. If the equation −5 csc 2 x + 12 cot 2 x − 9 = 0 is simplified using the identity 1 + cot 2 x = csc 2 x , the resulting equation is A. B. C. D. 18. −5 tan 2 x + 12 sec 2 x − 9 = 0 cot 2 x = 2 12 cot 2 x − 9 − 5sec 2 x = 0 sec 2 x (1 − tan 2 x ) = 6 The expression cos ( x − y ) − cos ( x + y ) is equivalent to A. 2 sin x sin y B. 0 C. −2 cos y ⎛ x− y⎞ D. cos ⎜ ⎟ ⎝ x+ y⎠ 19. 1 intersects the graph of cos 2 x − sin x twice in the interval 2 0 ≤ x < 2π . An equation that can be used to solve for x is The line y = A. B. C. D. cos 2 x = sin x 2 cos 2 x − 2 sin x − 1 = 0 sin x − cos 2 x = 2 2 cos 2 x + 2 sin x − 1 = 0 Principles of Math 12 - Trigonometry II Practice Exam 10 www.math12.com 20. 21. ⎛θ ⎞ ⎛ 2θ The expression sin ⎜ ⎟ cos ⎜ ⎝5⎠ ⎝7 ⎛ 17θ ⎞ A. cos ⎜ ⎟ ⎝ 35 ⎠ ⎛ 17θ ⎞ B. sin ⎜ ⎟ ⎝ 35 ⎠ ⎛ 3θ ⎞ C. sin ⎜ ⎟ ⎝ 35 ⎠ ⎛ −3θ ⎞ D. sin ⎜ ⎟ ⎝ 35 ⎠ If A. B. C. D. ⎞ ⎛ θ ⎞ ⎛ 2θ ⎞ ⎟ − cos ⎜ ⎟ sin ⎜ ⎟ is equivalent to ⎠ ⎝5⎠ ⎝ 7 ⎠ csc 2 x = 5 , then the value of x, to the nearest hundredth of a radian is sec 2 x 1.15 + 3.14n , n ∈ I 0.42 + 3.14n , n ∈ I 0.54 + 1.57 n , n ∈ I 0.21 + 1.57 n , n ∈ I Use the following information to answer the next question. A student is given four different trigonometric expressions I II III IV 1 sec x cos x 9 cot 2 x − csc2 x 2cos 2 x + 2sin 2 x 2 cos x 2 cot x sin x Numerical Response 5. If the expressions are simplified are ranked, from smallest to largest, the correct order is ________. Principles of Math 12 - Trigonometry II Practice Exam 11 www.math12.com 22. Given sin x = m , an expression for cos 2 x , in terms of m, is A. B. C. D. 23. 1 − 2m 2 1 − 2m 2m 2 − 1 2m − 1 The expression A. csc x + sin x sin x + 1 B. cos 2 x + 1 sin x + 1 C. sin 2 x D. 1 24. 1 + csc x is equivalent to sin x Given x = 450 , an equivalent expression to cos ( x + y ) is cos y ⎛x⎞ A. cos ⎜ ⎟ + 1 ⎝ y⎠ 2 B. (1 − tan y ) 2 C. cos x 2 + 2 cos y D. 2 cos y 25. ⎛ π⎞ The exact value of sec ⎜ − ⎟ is ⎝ 12 ⎠ 4 A. 2− 6 B. 6− 2 C. −750 11π D. 12 Principles of Math 12 - Trigonometry II Practice Exam 12 www.math12.com Numerical Response 6. 26. The expression cos x may be written as cos 2 kx − sin 2 kx . The value of k, to the nearest tenth, is ________. Using the identity cos 2 x = 1 − sin 2 x , the expression cos 2 x − sin 2 x − 1 + 2 sin x can be simplified to A. 2sin x (1 − sin x ) B. sin x (1 − 2sin x ) C. sin 2 x + 2sin x D. 2 sin 2 x + 1 27. 6 2 and sin y = − , the exact value of sec ( x + y ) , given that 7 5 3π 3π ≤ x < 2π , ≤ y < 2π , is 2 2 If tan x = − A. 21 − 5 B. 5 − 21 7 21 C. 12 85 − 5 D. 28. 5 85 7 21 − 12 The expression csc x − sin x is equivalent to 1 sin 2 x B. 1 sin x C. cos 2 x D. cot x cos x A. Principles of Math 12 - Trigonometry II Practice Exam 13 www.math12.com Numerical Response 7. 29. The number of solutions in the equation tan 2 x = 1 , where 0 ≤ x < 2π , is _________. The expression sin x + tan x is equivalent to cos x + 1 A. csc 2 x B. tan x 2 sin x C. cos x + 1 D. 2 tan x 30. The expression csc 4 x − 1 is equivalent to csc4 x A. sec4 x B. cot 4 x C. cot 2 x ( csc 2 x + 1) D. cot 2 x ( sec 2 x + 1) 31. The general solution to the equation sin 4 x = − 7π nπ ± 24 2 5π nπ B. , ± 12 4 7π nπ C. , ± 24 2 3π nπ D. ± 12 4 1 is 2 A. 3π nπ ± 12 4 11π nπ ± 24 2 Principles of Math 12 - Trigonometry II Practice Exam 14 www.math12.com 32. The expression sec 2 x is undefined when x is the angle A. B. π 4 π 2 C. π D. 2π Use the following information to answer the next question. A student solves the equation cos 2 x − 2 = 0 algebraically, using the steps shown below ( cos x − 2 ) ( cos x + 2 ) = 0 cos x − 2 = 0 cos x + 2 = 0 33. → x has no solution. → x has no solution. The reason why cos 2 x − 2 = 0 has no solution is because A. B. C. D. cos x is undefined for x = 2 ± 2nπ The range of y = cos x is −1 ≤ y ≤ 1 cos 2 x − 2 = 0 cannot be factored cos 2 x must be replaced with sin 2 x − 1 before factoring Principles of Math 12 - Trigonometry II Practice Exam 15 www.math12.com Use the following information to answer the next question. A student graphs the following function in a graphing calculator. f ( x ) = 8 − 3sin 2 x x is measured in radians, and the student wishes to analyze the graph for −2π ≤ x ≤ 2π Written Response – 10% • Explain how the student would have to type the above equation into their graphing calculator in order to obtain the correct graph. Indicate appropriate window settings. • 1. The student now wishes to solve the equation 6.2 = f ( x ) . State the general solution to this equation in radian decimal form, to the nearest hundredth. • The graph of f ( x ) can be expressed in the form g ( x ) = a cos b [ x − c ] + d . Write the equation for g ( x ) • Algebraically solve the equation 7 + sin 2 x = 8 − 3sin 2 x Show all steps required in obtaining the answer. Principles of Math 12 - Trigonometry II Practice Exam 16 www.math12.com Written Response – 10% 2. • Verify the identity cos x 1 + sin x π = for x = 1 − sin x cos x 6 Use the following additional information to answer the next part of the question. The graphs of y1 = • cos x 1 + sin x and y2 = are shown below. 1 − sin x cos x The graph of f ( x ) can be expressed in the form y1 = a sin b [ x − c ] + d or y2 = a cos b [ x − c ] + d . Write the equations for both y1 and y2. • cos x 1 + sin x and y2 = are not 1 − sin x cos x identical. Explain the difference between the graphs of y1 and y2. The graphs of y1 = Principles of Math 12 - Trigonometry II Practice Exam 17 www.math12.com • Algebraically prove the identity • Algebraically show that cos x 1 + sin x = 1 − si n x cos x cos x 1 + sin x 2 cos x + = 1 − sin x cos x 1 − sin x Principles of Math 12 - Trigonometry II Practice Exam 18 www.math12.com Written Response – 10% • Prove the identity 1 + cos 2 x = cot x sin 2 x • Prove the identity (sin x + cos x) 2 = 1 + sin 2 x • 3. Prove the identity sin 2 x = 2 sin x cos x Principles of Math 12 - Trigonometry II Practice Exam 19 www.math12.com • Solve algebraically: 2 sin x cos x = cos x • Solve algebraically: sin x sin x = 2 3 • Solve algebraically: csc x csc x 16 + = 5 3 15 Principles of Math 12 - Trigonometry II Practice Exam 20 www.math12.com ...
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This note was uploaded on 11/05/2011 for the course MATH 24325456 taught by Professor Jack during the Spring '09 term at Adventista de las Antillas.

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