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MA140 Sample Final Name: 20-point problems. Show all your work. Circle your answers. Sufficient work must be shown to receive any credit. 1. If tan x = − 43 with 0 < x < π, find the exact value of tan (x + π). 2. If tan x = − 43 with 0 < x < π, find the exact value of cot x. 3. If tan x = − 43 with 0 < x < π, find the exact value of sin x. 4. Given that A = 25° , a = 15, and b = 33, find the measure of angle B to the nearest degree. If there are two answers, give both of them. If there are no possible answers, write “none”. 1 5. Suppose sin x = 1 3 with π 2 < x < π. Find the exact value of sin 2x. 6. Find all exact solutions of the equation √ 2 sec x + 2 = 0 on the interval [0, 2π]. 7. Suppose that a supporting cable runs from the top of a 25 foot antenna to a point on the ground 20 feet from the base of the antenna. What is the angle between the cable and the ground (to the nearest degree)? 2 8. Referring to the diagram below, find the exact value of sin θ. 9. Referring to the diagram in problem #8, find the exact length of arc s cut off by the angle θ. 5 10. Suppose sin x = − 13 with sin(x + y). 3π 2 < x < 2π and cos y = − 53 with π < y < 3π . 2 Find the exact value of 11. Given that B = 110° , C = 39° , and b = 42 cm, find a to the nearest centimeter. If there are two answers, give both of them. If there are no possible answers, write “none”. 3 csc x − sin x = cos x (Hint: it is one of the six trigonometric functions: sin x, cos x, tan x, cot x, sec x, csc x) 12. Fill in the blank: 30-point problems. Show all work. Circle your answers. 13. Suppose tan x = − 34 with π < x < 2π. Find the exact value of cos x2 . 14. To find the length AB of a small lake, a surveyor at point C measures angle ACB to be 115°, length AC to be 500 feet, and length BC to be 325 feet. What is the length of the lake (to the nearest foot)? Circle your answer. 4 15. Find the amplitude, period, phase shift, and vertical shift of the function y = sin x3 − π3 . Also, graph the function over an interval of at least one period in length. Be sure to provide sufficient labels on your graph. Amplitude: Period: Phase Shift: Vertical Shift: 16. Verify the identity given below, using algebraic manipulation and trigonometric identities. Be sure to show all steps of your solution. Graphical solutions are not allowed for this problem. 1 + cos x sin x + = 2 csc x 1 + cos x sin x 5 17. Find all exact solutions for θ in degrees, 0° ≤ θ < 360°, of the equation: 2 cos2 θ = 1 − sin θ 18. If a force is applied using a rope with 100 lb tension at 24° above horizontal, find the horizontal and vertical components of the force (to the nearest pound.) Vertical: Horizontal: 19. Derive the sum of angles formulas for sine and cosine using Euler’s identity. 6