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Review Problems Quarterly #3 Exam Pre-Calculus 2017 1. Solve for all possible values of x within the domain of . Do not use your graphing calculator. a) b) c) d) e) f) 2. A ladder 3.0 m long is placed against a building at an angle of 60º with the ground. How high up the wall will the ladder reach? 3. To find the height of a pole, a surveyor moves 100 feet from the base of the pole and then, from an eyelevel height of 6.5 feet, measures the angle of elevation to the top of the pole to be 40º. Find the height of the pole to the nearest foot. 4. 5. The cable supporting a ski lift rises 3 feet for every 8 feet of horizontal length. The top of the cable is fastened 675 feet above the cable’s lowest point. Find the lengths b and c, and find the measure of Evaluate each: a.) b.) c.) d.) arccos(-.7398) e.) arctan(1.5) f.) arcsin(2.3) g.) h.) . 6. Find the amplitude, period, phase shift, and vertical shift of each function. a.) 7. b.) c.) Amp = _________ Amp = _________ Amp = _________ Period = ________ Period = ________ Period = ________ VS = __________ VS = __________ VS = __________ PS = __________ PS = __________ PS = __________ A windmill has a radius of 27 ft and its center is 52 ft above the ground. The wheel rotates at a constant angular velocity of radians per minute. The height of a point on the windmill’s blade as a function of time is given by the equation: , where h is the height in feet and t is the time in minutes. Find the approximate value of h(t) when t = 7 minutes. Chapter 5: Analytic Trigonometry Fundamental Identities Pythagorean Identities 2 2 sin x + cos x = 1 tan2 x + 1 = sec2 x 1 + cot2 x = csc2 x Reciprocal Identities 1 csc x = sin x 1 sec x = cos x 1 cot x = tan x Quotient Identities sin x cos x tan x = cot x = cos x sin x Identities for Negatives sin (-x) = - sin x cos (-x) = cos x tan (-x) = - tan x 8. Simplify: a.) sec x • cos x sin 2 x + cos x b.) cos x c.) cos u + sin u • tan u d.) sin t cos2 t −1 e.) csc x cot x + tan x f.) sin x 1 + cos x − 1 − cos x sin x Verifying Trigonometric Identities Hints: 1.) Work on one side of the equation only. Choose the more complicated side. 2.) If you have the sum or difference of fractions, try finding a common denominator. 3.) Try algebraically manipulating one side (i.e. factor, separate fractions, multiply, etc.). Verify each identity. 9. (sec x)(sin x + cos x) = tan x + 1 1− (cos x − sin x)2 = 2sin x cos x 10. cos A − sin A = csc A − sec A 11. sin A cos A 3cos 2 A + 5sin A − 5 3sin A − 2 = cos 2 A 1+ sin A 12. Sum, Difference, and Cofunction Identities sin (x + y) = sin x cos y + cos x sin y sin(x – y) = sin x cos y – cos x sin y cos (x + y) = cos x cos y – sin x sin y cos(x – y) = cos x cos y + sin x sin y tan x + tan y tan (x + y) = 1− tan x tan y tan x − tan y tan (x – y) = 1+ tan x tan y sin(90° – x) = cos x cos (90° – x) = sin x tan (90° – x) = cot x cot (90° – x) = tan x sec (90° – x) = csc x csc (90° – x) = sec x Evaluate exactly using Sum and Difference Identities. 13. sin 75° 14. cos74° cos44° + sin74° sin 44° 15. Evaluate cos 105° π 5 16. π 1 + tan π tan 5 Verify: 17. cot x − tan y = cos(x + y) sin x cos y tan π − tan 18. tan A + tan B = sin(A + B) tan A − tan B sin(A − B) Double-Angle Identities sin 2x = 2 sin x cos x 2 2 cos 2x = cos x – sin x = 1 – 2sin2 x = 2 cos2 x – 1 tan 2x = 2 tan x 1− tan 2 x Verify: 2 tan x 19. = sec 2 x sin 2x 20. (sin x – cos x)2 = 1 – sin 2x cos 2t 1+ tant = 21. 1− sin 2t 1− tant Trigonometric Equations Steps to Solve Trigonometric Equations: 1.) If only one trig function exists, isolate it and solve for the variable. 2.) If more than one trig function exists, try to factor out a common factor or substitute with identities. 3.) If two or more different trig. functions exist, try to use identities to rewrite in terms of one trigonometric function. 4.) If variables have different coefficients, try rewriting as the same variable (i.e. If sin 2x = sin x, rewrite sin 2x as 2sin x • cos x) 5.) Solve by factoring or by quadratic formula. Find exact solutions over the indicated interval. 22. cos x = − 3 2 0 ≤ x ≤ 360! 24. 2sin 2x = 3 1 cos 2 x = sin 2x 2 26. Estimate the solutions. 28. tan x = -0.84 0<x<p 0 ≤ x ≤ 360! −π π ≤x≤ 2 2 23. sin x = 1 2 all reals 25. 2 cos2 x + cos x = 1 27. 6sin2 x + 5 sin x = 6 0 < x < 2p 0° < x < 90°