Plane Trigonometry - Fall 1996 Test File Test #1 1.) Fill in the blanks in the two tables with the EXACT values of the given trigonometric functions. The total point value for the tables is 10 points. For each box that is NOT correct you will lose 1 point. Θ sin Θ cos Θ tan Θ cot Θ sec Θ csc Θ 0 30 45 60 90 Θ sin Θ cos Θ 135 240 480 π/3 5π/6 8π/3 11π/6 7π/4 2.) Prove that cos 2 A + sin 2 A = 1. Draw a picture of the triangle you use so I will know what the letters you use mean. 3.) Convert 24.38 to degrees/minutes/seconds. 4.) Convert 24 12 19 to decimal degrees. 5.) Given that tan A = -5/12 and A is not in Quadrant 2, find the values of the other trigonometric functions. Give exact answers.
6.) If cos A = k and 0 < A < 90, then what is cos(270 -A)? 7.) Convert the following to radian measure. a.) 24 b.) 2π 8.) Convert the following to degree measure. a.) 7π/5 b.) 45
9.) A woman is standing on the ground looking at a billboard. The billboard is on top of a building. The woman is 40 feet away from the building. The angle of elevation of the woman's line of sight to the top of the building is 67. The angle of elevation of the woman's line of sight to the top of the billboard is 76. How tall is the building? 10.) The wheel of a bicycle has diameter 24 inches. If the wheel is making 2 rev/sec, find the angular and linear velocity. 11.) For each of the following functions, list the period and the amplitude. If the amplitude does not exist, write "DNE". a.) y = 2 + 3 sin(2x - π/3) b.) y = - sin(2x + π) c.) y = cos(3x) 12.) Graph one complete period of the following functions. Label all of the important angles we talked about. Use radians. a.) y = -1 + 2 sin(x - π/3) b.) y = 3-2 cos 3x - π 3 13.) Graph the following function. Label the important angles we talked about. y = sec(x/6) Test #2 1.) Prove the following identities. a.) (sin A + cos A) 2 = 1 + sin 2A b.) sin 2Θ (tan Θ + cot Θ) = 2 c.) (1 + cos Θ)(cos Θ - 1) = -sin 2 Θ d.) 1 + 1 = 2 sec 2 A 1 + sin A 1 - sin A 2.) Rewrite the following function as a single trigonometric function and graph the resulting function. y = 2 sin x + 2 cos x 3.) Evaluate the following expressions. Give exact answers. a.) cos Sin -1-3 4 - Tan-1 2 7 b.) cos 22.5 c.) sin 52.5 cos 7.5 d.) Cos - 1 (cos(4π/3)) 4.) Find all x, with 0 x < 2π, that satisfy the following equations. Use radians. ANSWERS MUST BE EXACT, NOT APPROXIMATIONS!! a.) 1 + cos 2 x = sin x b.) sin x cos x = cos x c.) sin(2x) = 0 Test #3 1.) a.) If a = 22.1, c = 17.5 and B = 109, find A, C, b and the area of the triangle. b.) If a = 3, b = 4 and c = 6, find A, B, C and the area of
the triangle. c.) If A = 38.4, B = 54.1 and a = 14, find C, b, c and the area of the triangle. d.) If a = 8, b = 5 and A = 31, find c, B and C for any possible triangles. 2.) Do the following. a.) Graph r = 2 cos 3Θ. b.) Graph r = Θ cos Θ 3.) a.) Change the point (2, -2) to polar coordinates. b.) Change the point (4, 5π/6) to rectangular coordinates. c.) Change the point (-1, 5π/6) to rectangular coordinates. 4.) A man and a horse are pulling a block of granite. The force vector of the horse's pulling is <14, 9>. The man's force vector is <1, 2>. Find the magnitude of the resultant force. 5.) Let u = 2i - 3j, v = i + j and w = 3i. Find the following. a.) 2u - 3v b.) a unit vector in the same direction as v c.) u v d.) the angle between v and w e.) the projection of u in the direction of v f.) u Final Exam 1.) Prove the following identities. a.) (1 + cos Θ)(cos Θ - 1) = -sin 2 Θ b.) (sin A - cos A)(sin A - cos A) + 2 sin A cos A = 1 c.) tan 2 A csc 2 A = sec 2 A d.) sin 2Θ (tan Θ + cot Θ) = 2 3.) Prove that cos 2 A + sin 2 A = 1. Draw a picture of the triangle you use so I will know what the letters you use mean. 4.) Prove or disprove the following. Don't just say something like "They don't look the same." sin 2 θ + cos 2 θ = (sin θ + cos θ) 2 5.) Suppose that sec A = 4/3 and cot A < 0. Find the values of the other trigonometric functions. 6.) A woman is standing on the ground looking at a billboard. The billboard is on top of a building. The woman is 50 feet away from the building. The angle of elevation of the woman's line of sight to the top of the building is 57. The angle of elevation of the woman's line of sight to the top of the billboard is 61. How tall is the billboard? 7.) For each of the following functions, list the period and the amplitude. If the amplitude does not exist, write "DNE". a.) f(x) = 2 + 3 sin(4x - π/5) b.) f(x) = 2cot(x/3) 8.) Convert the following to radian measure. a.) 60 b.) π 9.) Convert the following to degree measure.
a.) π/3 b.) 45 10.) True - False - Circle the correct answer. cos 45 =.7071 sin(a + B) = sin A + sin B sin(2a) = 2 sin A cos(-θ) = cos Θ sec 2 x + 1 = tan 2 x 11.) Find the exact value for cos 75 - cos 15. 11.) Find the exact value for sin 75. 11.) Find the exact value for cos Sin -1-3 4 - Tan-1 2 7 14.) Rewrite the following function as a single trigonometric function and graph the resulting function. y = 2 sin x + 2 cos x 15.) Find all x, with 0 x < 2π, that satisfy the following equation. Use radians. ANSWER MUST BE EXACT, NOT AN APPROXIMATION!! cos 3x = 1/2 16.) Find all x, with 0 x < 2π, that satisfy the following equation. Use radians. ANSWER MUST BE EXACT, NOT AN APPROXIMATION!! 2 sin 2 x = 3 sin x 17.) Fill in the blanks. Give exact values, not approximations. Θ sin Θ cos Θ tan Θ 210 π/3 arcsin 3/7 18.) Write the following in standard form. a.) i -993 b.) (2 + 3i) + (-1 + 4i) c.) (2 + 3i)(-1 + 4i) d.) 2 + 3i -1 + 4i e.) i 944413 19.) a.) Change the point (2, -2) to polar coordinates. b.) Change the point (-1, 5π/6) to rectangular coordinates. 20.) Graph one period of the following function. List all of the important angles we talked about. Also put appropriate markings on the y-axis. y = 2-3 sin 2x - π 4