Alg2 - CH13 Practice Test

Transcription

1 lg - H13 Practice Test Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Find the value of the sine, cosine, and tangent functions for θ where = 96, = 8, and = 100. a. sin θ = 4 cos θ = 7 tan θ = 4 b. sin θ = 4 cos θ = 7 tan θ = 7 c. sin θ = 7 cos θ = 4 tan θ = 7 d. sin θ = 7 cos θ = 4 tan θ = 4. Use a trigonometric function to find the value of x. a. x = 50 c. x = 50 3 b. x = 45 3 d. x = 5 3. fter takeoff from an airport, an airplane s angle of ascent is 10. The airplane climbs to an altitude of 10,000 feet. t that point, what is the land distance between the airplane and the airport? Round your answer to the nearest foot. a. 1,763 ft c. 56,713 ft b. 57,588 ft d. 15,44 ft 4. raw an angle with 515 in standard position.

2 a. c. b. d. 5. Find the measure of the reference angle for θ = 14. a. 18 c. 38 b. 38 d P( 7, ) is a point on the terminal side of θ in standard position. Find the exact value of the six trigonometric functions for θ. a. sin θ = csc θ = cos θ = 7 sec θ = 7 tan θ = 7 cot θ = 7 b. sin θ = 7 csc θ = 7 cos θ = sec θ = tan θ = 7 cot θ = 7 c. sin θ = 7 csc θ = 7 cos θ = sec θ = tan θ = 7 cot θ = 7 d. sin θ = 7 csc θ = 7 cos θ = sec θ = tan θ = 7 cot θ = 7 7. Use the unit circle to find the exact value of the trigonometric function cos 30. a. 1 c. 1 b. 3 d.

3 8. Find all possible values of sin 1 3. a. π 3 + ( π)n, π 3 + ( π)n c. π 6 + ( π)n, 5π 6 + ( π)n b d. π 4 + ( π)n, 3π 4 + ( π)n 9. Solve the equation sin θ = 0.3 to the nearest tenth. Use the restrictions 90 < θ < 180. a. θ = 17.5 c. θ = 16.5 b. θ = d. θ = You can use trigonometry to measure the height of a pyramid in Egypt. [1.] n archaeologist positions himself 60 ft from the base of a pyramid so that his eye level is 5 ft above the ground. If the pyramid is 500 feet in height, what would be the angle of elevation from the archaeologist to the top of the pyramid? [.] The angle of elevation from the eye level of an archaeologist to the top of a pyramid whose base is 400 feet away is 50. To the nearest foot, what is the height of the pyramid? a. [1.] θ 6 c. [1.] θ 6 [.] h 48 ft [.] h 336 ft b. [1.] θ 63 d. [1.] θ 63 [.] h 477 ft [.] h 341 ft 11. triangle has a side with length 6 feet and another side with length 8 feet. The angle between the sides measures 73º. Find the area of the triangle. Round your answer to the nearest tenth. a. 3.0 ft c ft b ft d. 7.0 ft 1. Solve the triangle. m N = 118, m P = 33, and m = 15. Round to the nearest tenth. a. m M = 9, n 8., p 16.9 c. m M = 9, n 8., p 13.4 b. m M = 9, n 7.3, p 16.9 d. m M = 9, n 7.3, p 13.4 Numeric Response

4 13. If tan θ = 6 8, what is sinθ? 14. new kind of computer chip is made in the shape of a triangle. The distance of each side is shown in the diagram. Find the area of the new chip. Round to the nearest hundredth. Matching Match each vocabulary term with its definition. a. sine b. cosine c. Pythagorean Theorem d. tangent e. trigonometric function f. special triangle g. cotangent h. secant i. cosecant 15. in a right triangle, the ratio of the length of the leg opposite the angle to the length of the hypotenuse 16. the reciprocal of the sine function, or the ratio of the length of the hypotenuse to the length of the leg opposite the angle in a right triangle 17. the reciprocal of the cosine function, or the ratio of the length of the hypotenuse to the length of the leg adjacent to the angle in a right triangle 18. in a right triangle, the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse 19. the reciprocal of the tangent function, or the ratio of the length of the leg adjacent to the angle to the length of the leg opposite the angle in a right triangle 0. a function whose rule is given by a trigonometric ratio Match each vocabulary term with its definition. a. angle of rotation b. vertex angle c. initial side d. vector

5 e. standard position f. terminal side g. coterminal angle h. reference angle 1. for an angle in standard position, the positive acute angle formed by the terminal side of the angle and the x-axis. the initial position of a rotated ray 3. the terminal position of a rotated ray 4. an angle formed by a rotating ray and a stationary reference ray 5. an angle in standard position with the same terminal side 6. an angle whose vertex is at the origin and whose initial side is on the x-axis

6 lg - H13 Practice Test nswer Section MULTIPLE HOIE 1. NS: sin θ = opp. hyp. = = 7 cos θ = adj. hyp. = = 4 tan θ = opp. adj. = 8 96 = 7 The sine is the ratio of the length of the opposite leg to the length of the hypotenuse. The cosine is the ratio of the length of the adjacent leg to the length of the hypotenuse. orrect! The tangent is the ratio of the length of the opposite leg to the length of the adjacent leg. PTS: 1 IF: asic REF: Page 99 OJ: Finding Trigonometric Ratios TOP: 13-1 Right-ngle Trigonometry. NS: sin θ = opp. hyp. sin 45 = x 50 = x 50 NT: 1..1.m You know the length of the hypotenuse and want to find the length of the side opposite the given angle. Use the sine function. Substitute 45 for θ, x for opp., and 50 for hyp. Substitute for sin 45. x = 5 Multiply both sides by 50 to solve for x. The sine of angle is the ratio of the length of the opposite leg to the length of the hypotenuse. Use the sine function. Use the sine function. orrect! PTS: 1 IF: verage REF: Page 930 OJ: Finding Side Lengths of Special Right Triangles TOP: 13-1 Right-ngle Trigonometry 3. NS: NT: 1..1.m

7 tan θ = opp. adj. 10, 000 tan 10 = x Substitute 10 for θ, 10,000 for opp., and x for adj. x(tan 10 ) = 10,000 Multiply both sides by x. 10, 000 x = tan 10 56,713 ivide both sides by tan 10. Use a calculator to simplify. The land distance from the airplane to the airport is about 56,713 feet. The tangent function is the ratio of the length of the opposite leg to the length of the adjacent leg, not the length of the adjacent leg to the length of the opposite leg. Use the tangent function, not the sine function. orrect! Set your graphing calculator to interpret angle values as degrees, not radians. PTS: 1 IF: verage REF: Page 930 OJ: pplication NT: 1..1.m TOP: 13-1 Right-ngle Trigonometry 4. NS: Start with the initial side on the positive x-axis and rotate the terminal side 515 counterclockwise. Start with the initial side on the positive x-axis not the negative x-axis. Start with the initial side on the positive x-axis not the positive y-axis. Start with the initial side on the positive x-axis and then rotate the angle counterclockwise, not clockwise. orrect! PTS: 1 IF: verage REF: Page 936 OJ: rawing ngles in Standard Position TOP: 13- ngles of Rotation 5. NS: The reference angle is the acute angle created by the terminal side of θ and the x-axis. For example: When θ = 105, the reference angle measures 75.

8 When θ = 105, the reference angle also measures 75. The reference angle is the acute angle between the terminal side of the angle and the x-axis, not the y-axis. The reference angle is always positive. orrect! The reference angle is an acute angle PTS: 1 IF: asic REF: Page 937 OJ: Finding Reference ngles TOP: 13- ngles of Rotation 6. NS: Step 1 Plot point P, and use it to sketch angle θ in standard position. Find r.

9 r = ( 7) + ( ) = Step Find sin θ, cos θ, and tan θ. sin θ = y r = = cos θ = x r = 7 = 7 tan θ = y x = 7 = 7 Step 3 Use reciprocals to find csc θ, sec θ, and cot θ. csc θ = 1 sin θ = sec θ = 1 cos θ = 7 cot θ = 1 tan θ = 7 orrect! Plot the point P and sketch the angle in standard position. Plot the point P and sketch the angle in standard position. Plot the point P and sketch the angle in standard position. PTS: 1 IF: verage REF: Page 938 OJ: Finding Values of Trigonometric Functions 7. NS: The angle passes through the point cos θ = x cos 30 = 3 Ê Ë Á 3 1, ˆ on the unit circle. TOP: 13- ngles of Rotation The cosine of an angle is equal to the x-value of the point where the angle intersects the unit circle. orrect! The cosine of an angle is equal to the x-value of the point where the angle intersects the unit circle, not the y-value. The cosine of an angle is equal to the x-value of the point where the angle intersects the unit circle.

10 PTS: 1 IF: verage REF: Page 944 OJ: Using the Unit ircle to Evaluate Trigonometric Functions TOP: 13-3 The Unit ircle 8. NS: Step 1 Find the values between 0 and π radians for which sin θ is equal to 3. 3 = sin π 3, 3 = sin π 3 Use y-coordinates of points on the unit circle. Step Find the angles that are coterminal with angles measuring π 3 and π 3 radians. π 3 + ( π)n, π 3 + ( π)n dd integer multiples of π radians, where n is an integer. orrect! Find the possible values of the inverse of the sine function, not the reciprocal of the sine function. Find the angles on the unit circle where the y-value is the square root of 3 over, then add integer multiples of pi. Find the angles on the unit circle where the y-value is the square root of 3 over, then add integer multiples of pi. PTS: 1 IF: verage REF: Page 950 OJ: Finding Trigonometric Inverses TOP: 13-4 Inverses of Trigonometric Functions 9. NS: Using the inverse sine function we get θ = sin 1 (0.3) ecause 90 < θ < 180 we need to find the angle in Quadrant II that has the same sine value as θ = 16.5 heck the range given in the problem. Find the reference angle from the x-axis. orrect! heck the range given in the problem. PTS: 1 IF: verage REF: Page 95 OJ: Solving Trigonometric Equations TOP: 13-4 Inverses of Trigonometric Functions

11 10. NS: [1.] The angle of elevation, θ, may be found using the tangent function, tan θ = opposite adjacent. The opposite side is the distance from eye level to the top of the pyramid, or 495 feet. The adjacent side is the distance from the archaeologist to the base of the pyramid, or 60 ft. tan θ = θ = tan [.] The height of the pyramid, h, from the archaeologist s eye-level to the top, may be found using the tangent function, tan θ = h. The angle of elevation is 50. The adjacent side is the adjacent distance from the archaeologist to the base of the pyramid, or 400 ft. tan 50 = h 400 h = 400 tan ft dding 5 ft to this gives a total height of 48 ft. orrect! The tangent of an angle is the opposite side divided by the adjacent side. The tangent of an angle is the opposite side divided by the adjacent side. The tangent of an angle is the opposite side divided by the adjacent side. PTS: 1 IF: dvanced TOP: 13-4 Inverses of Trigonometric Functions 11. NS: area = 1 bh Write the area formula. h = asin Solve for h using trigonometric ratios. area = 1 absin area = 1 ft (6)(8) sin 73 = 3.0 Substitute. orrect! The first term of the area formula for a triangle is one half. The area of a triangle is one half the product of the lengths of two of the sides and the sine of their included angle.

12 The area of a triangle is one half the product of the lengths of two of the sides and the sine of their included angle. PTS: 1 IF: asic REF: Page 958 OJ: etermining the rea of a Triangle 1. NS: Step 1 Find the third angle measure. m M + m N + m P = 180 m M = 180 m M = 9 TOP: 13-5 The Law of Sines Step Find the unknown side lengths. sin M m = sin N sin M Law of Sines n m = sin P p sin 9 = sin 118 sin 9 Substitute. = sin n 15 p n = 15 sin 118 Solve for the unknown p = 15 sin 33 sin 9 side. sin 9 n 7.3 p 16.9 Use the Law of Sines correctly to find the measure of n. orrect! Use the Law of Sines correctly. Use the Law of Sines correctly to find the measure of p. PTS: 1 IF: verage REF: Page 959 OJ: Using the Law of Sines for S and S TOP: 13-5 The Law of Sines NUMERI RESPONSE 13. NS: 3 5 PTS: 1 IF: verage TOP: 13-1 Right-ngle Trigonometry 14. NS: PTS: 1 IF: dvanced TOP: 13-6 The Law of osines MTHING 15. NS: PTS: 1 IF: asic REF: Page 99 TOP: 13-1 Right-ngle Trigonometry

13 16. NS: I PTS: 1 IF: asic REF: Page 93 TOP: 13-1 Right-ngle Trigonometry 17. NS: H PTS: 1 IF: asic REF: Page 93 TOP: 13-1 Right-ngle Trigonometry 18. NS: PTS: 1 IF: asic REF: Page 99 TOP: 13-1 Right-ngle Trigonometry 19. NS: G PTS: 1 IF: asic REF: Page 93 TOP: 13-1 Right-ngle Trigonometry 0. NS: E PTS: 1 IF: asic REF: Page 99 TOP: 13-1 Right-ngle Trigonometry 1. NS: H PTS: 1 IF: asic REF: Page 937 TOP: 13- ngles of Rotation. NS: PTS: 1 IF: asic REF: Page 936 TOP: 13- ngles of Rotation 3. NS: F PTS: 1 IF: asic REF: Page 936 TOP: 13- ngles of Rotation 4. NS: PTS: 1 IF: asic REF: Page 936 TOP: 13- ngles of Rotation 5. NS: G PTS: 1 IF: asic REF: Page 937 TOP: 13- ngles of Rotation 6. NS: E PTS: 1 IF: asic REF: Page 936 TOP: 13- ngles of Rotation