768 CHAPTER 7 Trigonometric Identities and Equations Chapter 7 Test Work each problem. 1. If cos u = 24 25 and u is in quadrant IV, find the other five trigonometric functions of u. 2. Express sec u - sin u tan u as a single function of u. 3. Express tan2 x - sec2 x in terms of sin x and cos x, and simplify. 4. Find the exact value of cos 5p 12 . 5. Express (a) cos1270° - u2 and (b) tan1p + x2 as functions of u or x alone. 6. Use a half-angle identity to find the exact value of sin1-22.5°2. 7. Given that sin A = 5 13 , cos B = - 3 5 , A is a quadrant I angle, and B is a quadrant II angle, find each of the following. (a) sin1A + B2 (b) cos1A + B2 (c) tan1A - B2 (d) the quadrant of A + B 8. Given that cos u = - 3 5 and 90° 6u 6180°, find each of the following. (a) cos 2u (b) sin 2u (c) tan 2u (d) cos u 2 (e) tan u 2 135. Snell’s Law Refer to Exercise 134. What happens when u1 is greater than the critical angle? 136. British Nautical Mile The British nautical mile is defined as the length of a minute of arc of a meridian. Because Earth is flat at its poles, the nautical mile, in feet, is given by L = 6077 - 31 cos 2u, where u is the latitude in degrees. See the figure. (Data from Bushaw, D., et al., A Sourcebook of Applications of School Mathematics, National Council of Teachers of Mathematics.) Give answers to the nearest tenth if applicable. A nautical mile is the length on any of the meridians cut by a central angle of measure 1 minute. (a) Find the latitude between 0° and 90° at which the nautical mile is 6074 ft. (b) At what latitude between 0° and 180° is the nautical mile 6108 ft? (c) In the United States, the nautical mile is defined everywhere as 6080.2 ft. At what latitude between 0° and 90° does this agree with the British nautical mile? Verify that each equation is an identity. 9. sec2 B = 1 1 - sin2 B 10. cos 2A = cot A - tan A csc A sec A 11. tan2 x - sin2 x = 1tan x sin x22 12. tan x - cot x tan x + cot x = 2 sin2 x - 1
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