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Honors Math Analysis Fall Mid-Term Exam All numerical answers expressed to three places after the decimal (y.xxx). 1. The point (6, –12) is on the terminal side of an angle in standard position. Give the smallest positive angle in both degrees and radians. 12 1 12 tan tan 63.435 6 6 63.435 360 296.565 296.565 180 5.176 2. Give the exact values of the six trigonometric 19 functions of the angle . 6 19 12 7 7 7 5 2 2 6 6 6 6 6 6 5 1 19 19 sin ;csc sin 2 6 2 6 6 19 cos 6 19 tan 6 3 19 ;sec 2 6 2 3 1 19 ;cot 3 3 6 Honors Math Analysis Fall Mid-Term Exam 3. State the (a) amplitude, (b) period, (c) phase shift, (d) domain, and (e) range for the sinusoid f x 1.5sin 2 x . 4 f x 1.5sin 2 x 1.5sin 2 x 4 8 2 ; phase shift = 2 8 Domain : x : x ; Range : y : 1.5 y 1.5 amplitude = 1.5, period = 4. Write g x 2 cos 2 x 3sin 2 x as a sinusoid in the form y a sin bx c . y a sin bx c a sin bx cos c a cos bx sin c a cos x 2, a sin x 3, b 2 a 2 cos 2 x sin 2 x 4 9 13 a 13 c cos 1 2 56.310 13 g x 2 cos 2 x 3sin 2 x 13 sin 2 x 56.310 = 13 sin 2 x 0.93 Honors Math Analysis Fall Mid-Term Exam 5. Find the length of the arc intercepted by a central 5 angle of rad in a circle of radius 4. 4 5 s r 4 5 4 6. Dr. Thom Lawson standing on flat ground 62 ft from the base of a Douglas fir measures the angle of elevation to the top of the tree as 72º24′. What is the height of the tree? 24 h 7224 72 72.4 tan 72.4 60 62 h 62 tan 72.4 195.449 ft 7. What would be the angle of elevation of the top of a 150 ft building as measured from a point 78 ft from its base? 150 1 150 tan tan 62.526 78 78 8. If sin 0.843 and cos 0.537 , find sin 2θ. sin 2 2sin cos 2 0.843 0.537 0.905 Honors Math Analysis Fall Mid-Term Exam 2 9. Prove : sin 3 3cos sin sin 3 3cos 2 sin sin 3 sin 3 sin 2 sin 2 cos cos 2 sin 2sin cos cos cos 2 sin 2 sin 2sin cos 2 cos 2 sin sin 3 3cos 2 sin sin 3 10. Use your calculator to conjecture whether sec x sin x tan x cos x is likely to be an identity. Confirm your conjecture algebraically. sec x sin x tan x cos x is likely to be an identity. 1 sin x cos x sec x sin x tan x sin x cos x cos x 1 sin 2 x cos 2 x cos x cos x cos x Honors Math Analysis Fall Mid-Term Exam 11. Find and state all the solutions to cos3 x 2sin x 0.7 0 in the interval [0, 2π) graphically. cos3 x 2sin x 0.7 0 at x 0.137,3.789 12. Two markers A and B on the same side of a canyon are 80 ft apart, as shown in the figure. A hiker is located on the opposite rim at point C. A surveyor determines that BAC 70 and ABC 65. (a) What is the distance between the hiker and point A? C 180 70 65 45 b c sin B sin C c sin B 80sin 65 b 102.537 ft sin C sin 45 Honors Math Analysis Fall Mid-Term Exam (b) What is the distance between the two canyon rims? Assume they are parallel. d sin 70 102.537 d 102.537 sin 70 96.353 ft 13. To determine the distance between two points A and B on opposite sides of a lake, a surveyor chooses a point C that is 900 ft from A and 225 ft from B, as shown in the figure. If the measure of the angle at C is 70º, find the distance from A to B. c 2 a 2 b 2 2ab cos C c 225 900 2 225 900 cos 70 2 849.769 ft 2 Honors Math Analysis Fall Mid-Term Exam 14. Find the area of the triangle with sides 8, 9, 10 in length. 8 9 10 s 13.5 2 A 13.5 13.5 8 13.5 9 13.5 10 13.5 5.5 4.5 3.5 34.197 Honors Math Analysis Fall Mid-Term Exam 15. Extra Credit Building inspector Julie Wong checks a building in the shape of a regular octagon 20 ft on a side. She checks that the contractor has located the corners of the foundation correctly by measuring several diagonals. Calculate what the lengths of HB, HD, and HC should be. 360 180 45 45 67.5 8 2 HR 20 20sin 67.5 HR 26.131 sin 67.5 sin 45 sin 45 HB 26.131 26.131 36.955 ft 2 2 26.131sin135 48.284 ft sin 22.5 HD 2 HR 52.263 ft HC