(1) Trigonometry Test Answer Key

(1) Trigonometry Test Answer Key - Trigonometry I Copyright...

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Unformatted text preview: Trigonometry I Copyright © 2006, Barry Mabillard. Trigonometry I Standards Test Practice Exam - ANSWERS 0 www.math40s.com www.math40s.com 1. The minimum and the maximum of a trigonometric function are shown in the diagram. a) Write a cosine equation for the function First determine the parameters a, b, c, & d. 7 - (-3 ) max - min 10 = = =5 2 2 2 *The horizontal distance between the two points is 8 units, so a full cycle (the period) is 16 units. 2π 2π π b= = = P 16 8 c = 5 (that is where the cosine pattern begins) min+ max -3 + 7 4 d= = = =2 2 2 2 a= ⎡π ⎤ The equation is y = 5cos ⎢ ( x - 5 ) ⎥+ 2 ⎣8 ⎦ b) Determine the value of the y – intercept, correct to three decimal places Find the y-intercept by setting x = 0, then solving for y. ⎡π ⎤ y = 5cos ⎢ ( x - 5 ) ⎥+ 2 ⎣8 ⎦ ⎡π ⎤ Alternatively, graph in your y = 5cos ⎢ (0 - 5 ) ⎥+ 2 8 ⎣ ⎦ calculator and type 2nd → Trace → Value → x = 0 ⎡-5 π ⎤ y = 5cos ⎢ +2 ⎣8 ⎥ ⎦ y = 0.087 11π to degrees 18 11π 180 o × = 110 o 18 π 2. Convert Trigonometry I Standards Test Practice Exam - ANSWERS 1 www.math40s.com 1 , and the terminal arm is in Quadrant I. 7 adj First draw in the triangle, using the fact cosθ = hyp 3. Find the exact value of sinθ if cos θ = Use Pythagoras to find the unknown side: a 2 + b2 = c2 (1) 2 + b2 = ( 7) 2 1 + b2 = 7 b2 = 6 It follows that sinθ = b=± 6 *Reject the negative case since we know the terminal arm is in quadrant I. 6 7 4. What is the period in the function f ( x ) = 3sin ( 5 x ) − 2 The b –value is 5 2 π 2π The period is P = = b 5 5. Given the point (-3, -4), determine the exact values of cosθ and sinθ Draw the information you know, then use Pythagoras to solve for the unknown side: a 2 + b2 = c2 ( −3) + ( −4 ) 2 2 -3 5 -4 sinθ = 5 = c2 cosθ = 9 + 16 = c 2 25 = c 2 c=5 6. The length of arc swept out by an angle θ is 50 cm. If the radius of the circle is 18 cm, determine the measure of θ in radians. Use the arc length formula s = θr 50 cm= θ ( 18 cm ) θ = 2.78 rad Trigonometry I Standards Test Practice Exam - ANSWERS 2 www.math40s.com 7. Determine a value for x that would make the function f ( x ) = 3csc x undefined in the domain [0, 2π ) Recall that cscx = 1 , so look for values of x that make sinx = 0 sinx A possible answer is 0 rad or π rad. 2π rad may not be used because it is not included in the allowed domain for the question. 8. Determine the value of f (10 ) in the function f ( x ) = 4sin ⎡π ( x − 2 ) ⎤ + 3 ⎣ ⎦ f ( 10 ) = 4sin ⎡π ( 10 - 2 ) ⎤ + 3 ⎣ ⎦ f ( 10 ) = 4sin [8π ]+ 3 f ( 10 ) = 4 ( 0 ) + 3 f ( 10 ) = 3 3π ⎞ ⎛ 9. The graph of y = 6 cos ⎜ x + ⎟ + 1 is illustrated below. 4⎠ ⎝ Determine the exact values of g, h, m & n. From the function, a = 6. The b – value is 1. 3π (shifted left) phase shift = 4 d = 1. 3π 4 h: 1 + 6 = 7 g: − m: (Half a period past point g) 3π 3π 4π π − +π = − + = 4 4 4 4 n: 1 – 6 = -5 Trigonometry I Standards Test Practice Exam - ANSWERS 3 www.math40s.com 10. If the measure of the central angle is π 4 , determine the measures of the other two angles within the triangle. All angles in a triangle add up to 180°. (or π). In an isosceles triangle, the two angles marked by x are equivalent. It follows that: x + x + 45o = 180o 2 x = 135 o x = 67.5o Therefore, the measure of the angles is π 3π 67.5 o × = 0 180 8 11. Determine the number of radians between the hour hand and the minute hand at 7:00. 7 of the 12 way around the circle. 7:00 is exactly If 2π is the whole circle, then 2π× Trigonometry I Standards Test Practice Exam - ANSWERS 7 7π = 12 6 4 www.math40s.com ⎛ 7π ⎞ 12. Determine the value of cos ⎜ − ⎟ ⎝ 4⎠ - 7π π is a co-terminal angle with 4 4 2 ⎛π ⎞ cos ⎜ ⎟ = 2 ⎝4 ⎠ 13. The circle x 2 + y 2 = 1 is drawn below, along with the line y = 3 . 2 Determine the coordinates of the two intersection points. The points on the unit circle that have a 3 are located at y – coordinate of 2 π 2π and 3 3 ⎛1 3 ⎞ are ⎜ , ⎜2 2 ⎟ ⎟ 3 ⎝ ⎠ ⎛1 3 2π The coordinates at are ⎜- , ⎜2 2 3 ⎝ The coordinates at π ⎞ ⎟ ⎟ ⎠ 14. The graph of f ( x ) = 6sin x + d touches the x-axis once (but does not pass through) on the interval 0 ≤ x ≤ 2π . A possible value for d is: There are two possibilities: The graph could touch at the top The d-value here is -6 Trigonometry I Standards Test Practice Exam - ANSWERS or at the bottom. The d-value is +6 5 www.math40s.com 15. a) Graph y = cos x b) Graph y = cos −1 x c) State the domain of f −1 ( x ) { x | -1 ≤ x ≤ 1, x ∈ R} 16. The maximum point on a trigonometric graph is at the point (-4, 6), and the minimum point is at (2, -2). If the graph is of the form y = a cos ⎡b ( x + c ) ⎤ + d , then determine ⎣ ⎦ possible values for each of the parameters. 6 - (-2 ) max - min = = 2 2 * The period is 12 units 2π 2π π b= = = P 12 6 c=4 max + min 6 + (-2 ) d= = = 2 2 a= 8 =4 2 4 =2 2 ⎡π ⎤ The equation is y = 4cos ⎢ ( x + 4 ) ⎥+ 2 ⎣6 ⎦ Trigonometry I Standards Test Practice Exam - ANSWERS 6 www.math40s.com 17. An angle of 15° is equivalent to _______ radians. (exact value) 15 o × π 180 o = π 12 5π ⎞ ⎛ 18. The exact value of cos −1 ⎜ cos ⎟ is 6⎠ ⎝ You can do this using degree mode in your calculator: cos150 o = -0.8660 cos -1 (-0.8660 ) = 150 o or 5π 6 19. Write the general equation of a vertical asymptote in the graph of y = csc x 1 Rewrite csc x as . Vertical asymptotes occur when sin x = 0 sinx A vertical asymptote occurs when x = 0, π, 2π, and so on. The general solution would be { x | x = kπ , k ∈ I } 20. If csc θ = 2 and tan θ < 0 , determine the value of cos θ Since cscθ is positive, and tanθ is negative, the terminal arm must be in quadrant II Recall that cscθ = hyp 2 = opp 1 Use Pythagoras to determine the unknown side. a 2 + b2 = c2 a 2 + 12 = 22 a2 + 1 = 4 a2 = 3 a=− 3 Use a negative since the base of the triangle lies on the negative x-axis. It follows that cosθ = - 3 2 Trigonometry I Standards Test Practice Exam - ANSWERS 7 www.math40s.com ⎡π ⎤ 21. Given the function f ( x ) = 2sin ⎢ ( x − 2 ) ⎥ ⎣2 ⎦ a) Sketch the graph b) Sketch y = f ( x ) 22. If the coordinates of a point P (θ ) on the unit circle is (a, b), then the coordinates of the point P ⎡θ + 180o ⎤ are ⎣ ⎦ This question should be visualized on the unit circle: If you look at a point exactly 180° away from point P(a, b), you can see that both coordinates change sign. The coordinates of P ⎡θ + 180 o ⎤ are (-a, -b) ⎣ ⎦ Trigonometry I Standards Test Practice Exam - ANSWERS 8 www.math40s.com 23. State the period of the graph of y = csc θ The period of y = cscθ is 2π 24. Convert 3π to degrees. Express answer to one decimal place. 5 3 π 180 o × = 108 o π 5 25. Given the function f ( x ) = tan x ⎛ π π⎞ a) Sketch y = f ( x ) on the domain ⎜ − , ⎟ ⎝ 2 2⎠ b) State the domain of f ( x ) π ⎧ ⎫ ⎨ x | x ≠ + nπ , n ∈ I ⎬ 2 ⎩ ⎭ c) Sketch the graph of f −1 ( x ) Trigonometry I Standards Test Practice Exam - ANSWERS 9 www.math40s.com 26. A floating ball in a lake goes up and down with the tide. At 1 second, the ball has a minimum height of 5 cm below surface level. At 3 seconds, the ball has a maximum height of 5 cm above surface level. a) Sketch a graph for the first 4 seconds of motion b) Write an equation for the function a=5 2π 2π π b= = = 2 P 4 c=2 d=0 ⎛π ⎞ h (t ) = 5sin ⎜ (t - 2 ) ⎟ ⎝2 ⎠ ⎛π ⎞ OR h (t ) = -5sin ⎜ t ⎟ ⎝2 ⎠ 27. If the product of cos x and sin x is negative, then which quadrant is the angle in? This is true in the quadrants where x & y have different signs. The answer is quadrants II & IV ⎛1⎞ 28. The value of sin −1 ⎜ ⎟ is ⎝2⎠ In your calculator (using degree mode) evaluate sin-1(0.5) = 30° π⎞ ⎛ 29. The exact value of sin ⎜ cos ⎟ is 2⎠ ⎝ π ⎛ sin ⎜ cos 2 ⎝ (or π 6 ) ⎞ ⎟ = sin (0 ) = 0 ⎠ Trigonometry I Standards Test Practice Exam - ANSWERS 10 www.math40s.com ⎛1 3⎞ 30. If the coordinates of a point P (θ ) on the unit circle are ⎜ , − ⎟ , then the ⎜2 2⎟ ⎝ ⎠ o coordinates of the point P ⎡θ + 180 ⎤ are ⎣ ⎦ Based on the coordinates, the angle is 300°. If the point is rotated 180°, the new angle is 120°. ⎛ 1 3⎞ The coordinates are ⎜- , ⎜2 2⎟ ⎟ ⎝ ⎠ 31. A tire rolls 3π metres while turning 240°. Determine the area of the wheel. Use the arc length formula. (First convert the angle to radians!) s = θr ⎛ 4π ⎞ 3π = ⎜ ⎟r ⎝3⎠ 4π r 3π = 3 9π = 4π r 9 r= 4 Now use the formula for the area of a circle: A= π r 2 2 ⎛9 ⎞ A= π ⎜ ⎟ ⎝4 ⎠ A= 15.9 m2 Trigonometry I Standards Test Practice Exam - ANSWERS 11 www.math40s.com Multiple Choice Answers: 1. Given csc θ = 5 , the exact value of tan θ is 4 5 4 3 b) 5 First rewrite the equation as sinθ = 3 4 4 d) ± 3 Without any further information, the angle could be in quadrant I a) c) ± 4 5 or the angle could be in QII Both cases will give a positive value for sinθ Use Pythagoras to solve for the unknown sides. Based on these triangles, tanθ =± 4 3 The answer is d. Trigonometry I Standards Test Practice Exam - ANSWERS 12 www.math40s.com π m. The tire is rolled and travels a total distance of 28π m. 5 By the time the tire stops, it has rolled through an angle of 2. A tire has a radius of a) 28π b) 140° 28 c) rad 5 d) 8021.41° Use the arc length formula s = θr The arc length s is 28π m The radius is ⎛π ⎞ 28π = θ ⎜ ⎟ ⎝5 ⎠ θπ 28π = 5 140 π = θ π π 5 m θ = 140 This means the angle is 140 radians. Since this is not a solution, convert to degrees. 140 rad× 180 0 π = 8021.410 The answer is d. 3. Given the trigonometric function f ( x ) = cos x , the statement which is true is a) f ( x) = − f ( x) b) f ( x ) = f ( -x ) c) f ( x ) = f −1 ( x ) d) f ( x) = − f (−x) 1 05 . −2 π − 3 π 2 −π − π 2 π 2 π 3π 2 2π -. 05 1 Based on the above graph, it can be seen that a reflection about the y – axis will result in the exact same graph. The answer is b. Trigonometry I Standards Test Practice Exam - ANSWERS 13 www.math40s.com 4. If 900 < θ < 1800 , a true statement is a) 00 < θ < 900 The angle is in quadrant II. b) cos θ ≤ tan θ c) 0 < sinθ < 1 d) 0 < cos θ < 1 We know that sin90 0 = 0 and sin180 0 = 1 Therefore, 0 < sinθ < 1 The answer is c. 5. Given the function f ( x ) = 12 cos ( 2 x ) , and the transformation g ( x ) = then the amplitude of g ( x ) is a) b) c) d) 2 3 4 12 1 f ( x) , 4 1 f (x) . 4 Replace f ( x ) with the actual function Start with g ( x ) = 1 ⎡12cos ( 2x ) ⎤ ⎦ 4⎣ Now multiply the fraction into the brackets g ( x ) = 3cos ( 2x ) g (x)= The answer is b. Trigonometry I Standards Test Practice Exam - ANSWERS 14 www.math40s.com 6. The function f ( x ) = 2sec x has a range of a) ( −2, 2 ) b) (−∞, −2] ∪ [2, ∞) ⎡ 1 1⎤ c) ⎢ − , ⎥ ⎣ 2 2⎦ d) ( −∞, ∞ ) From the diagram, it can be seen that the graph exists beneath -2, and above 2. The answer is b. 7. A wheel turns through an angle of 16 radians. This angle measurement in degrees is a) b) π0 8 180 π c) 8π 0 d) 16 rad× 180 0 π = 2880 0 The answer is d. π 2800 0 π 8. A sine function has a range of [-6, 2] and a period of 4. A trigonometric equation with these properties is ⎛π ⎞ max - min 2- (-6 ) 8 a) y = 8 sin ⎜ θ ⎟ + 2 a= = = =4 2⎠ ⎝ 2 2 2 b) y = 4sin ( 4θ ) − 2 2π 2π π b= = = P 4 2 ⎛π ⎞ c) y = 4sin ⎜ θ ⎟ - 2 c=0 ⎝2 ⎠ min+ max -6+ 2 -4 ⎛2 ⎞ d= = = = -2 d) y = 4sin ⎜ θ ⎟ − 2 2 2 2 ⎝π ⎠ ⎛π ⎞ The equation is y = 4sin ⎜ θ ⎟ - 2 ⎝2 ⎠ The answer is c. Trigonometry I Standards Test Practice Exam - ANSWERS 15 www.math40s.com 9. The angle 24π is co-terminal to an angle of 3 a) 0˚ 2π b) 3 5π c) 6 7π d) 3 24π = 8π = 14400 3 1440 0 - 360 0 - 360 0 - 360 0 - 3600 = 00 The answer is a. ⎛ 7π ⎞ 10. The exact value of csc ⎜ ⎟ is ⎝4⎠ a) - 2 2 b) − 2 2 c) 2 d) 2 11. Given that cos θ = a) b) c) d) I II III IV 1 ⎛ 7π ⎞ csc ⎜ ⎟= ⎝ 4 ⎠ sin ⎛ 7 π ⎜ ⎝4 ⎞ ⎟ ⎠ = 1 2 =2 2 2 Rationalize the denominator: 2 2 -2 2 × = =- 2 2 2 2 The answer is a. 4 3 and sin θ = − , then the terminal arm is located in quadrant 5 5 A positive cosine means the terminal arm is located in quadrant I or IV A negative sine means the terminal arm is located in quadrant III or IV There is overlap in IV. The answer is d. 12. The period of the function g(x) is 8. If g ( 0 ) = 12, g ( 4 ) = 6, and g ( 8 ) = 12 , then the value of g (12 ) is a) b) c) d) 0 6 12 18 Draw in the values to see what is happening, then project the graph to the x-value of 12. The value of g ( 12 ) is 6. The answer is b. Trigonometry I Standards Test Practice Exam - ANSWERS 16 www.math40s.com 13. An angle of 13 , 22 22 b) − , 22 c) (0, -1) d) (-1, 0) a) 3π on the unit circle has coordinates of 2 The coordinates can be read off the unit circle, and are (0, -1) The answer is c. 14. If sec θ > 0 and sin θ < 0 , then θ terminates in quadrant a) I b) II A positive secant means the terminal c) III arm is located in quadrant I or IV d) IV A negative sine means the terminal arm is located in quadrant III or IV There is overlap in IV. The answer is d. 1 π⎞ ⎛ 15. The period and phase shift for the trigonometric equation y = sin 4 ⎜ θ − ⎟ are 2 3⎠ ⎝ π a) period = 4 ; phase shift = left 3 2π 2π π π = =. The period is b) period = 4 ; phase shift = right b 4 2 3 π π π units right. The phase shift is c) period = ; phase shift = left 3 2 3 The answer is d. π π d) period = ; phase shift = right 2 3 16. The equation of an asymptote on the graph of y = csc x is a) x = b) x = π 4 π 2 c) x = π 3π d) x = 2 The answer is c. Trigonometry I Standards Test Practice Exam - ANSWERS 17 www.math40s.com 17. If tan x sin x < 0 , then x terminates in quadrants a) II or III b) II or IV tanxsinx < 0 occurs when c) I or IV tanx and sinx have different d) III or IV signs. (Recall the product of a positive and negative is negative) They have different signs in quadrants II (tanx is negative, sin x is positive) and quadrant III (tanx is positive, sinx is negative.) The answer is a. ⎛ 2⎞ 18. The value of sin −1 ⎜ ⎟ ⎜ 2 ⎟ is: ⎝ ⎠ π a) 4 5π b) 4 4π c) 3 11π d) 6 When θ = π 4 , the value of sinθ is 2 2 ⎛ 2⎞ Alternatively, type in sin-1 ⎜ ⎟ ⎜ 2 ⎟ into ⎝ ⎠ your calculator in degree mode to get back 450 The answer is a. Trigonometry I Standards Test Practice Exam - ANSWERS 18 www.math40s.com ...
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This note was uploaded on 02/09/2011 for the course 168 comm 168 taught by Professor Kiscabean during the Winter '10 term at UCLA.

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