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Review TEST 2 MAC 1114 Sum B Name___________________________________ Solve the problem. 1) A wheel with a 23-inch diameter is turning at the rate of 47 revolutions per minute. To the nearest inch per minute, what is the velocity of a point on the rim? Answer: 3396 in./min Graph the function over a one-period interval. 1 2) y = 2 + sin (2x - ) 3 Answer: Find the exact value of the expression in degrees without using a calculator or table. 2 3) = cos-1 2 Answer: 45° 4) = csc-1 (-2) Answer: -30° 5) = sec-1 - 2 Answer: 135° 1 Find the exact value of the composition. 6) sin (arctan (2)) Answer: 2 5 5 7) cos arcsin 1 4 Answer: 15 4 3 8) tan-1 tan 4 Answer: - 4 Use a calculator to find the approximate value of the composition. Round answers to four decimal places. The expression may be undefined. 9) sin (cos-1 (0.8324)) Answer: 0.5542 Find the exact value of the indicated trigonometric function for the given right triangle. 10) 20 12 Find sin A and cos A. Answer: sin A = 4 3 ; cos A = 5 5 Solve the right triangle with the given sides and angles. 11) a = 3.8 cm, b = 1.4 cm Answer: = 69.8°, = 20.2°, c = 4.0 cm Perform the computation and give answer with the appropriate number of significant digits. 62 12) sin (21.597°) Answer: 170 Solve the problem. 13) From a boat on the lake, the angle of elevation to the top of a cliff is 33°7'. If the base of the cliff is 986 feet from the boat, how high is the cliff (to the nearest foot)? Answer: 643 ft 2 14) From a distance of 48 feet from the base of a building, the angle of elevation to the top of the building is 69°. Estimate the height of the building to the nearest foot. Answer: 125 ft 15) A blimp is 1700 meters high in the air and measures the angles of depression to two stadiums to the west of the blimp. If those measurements are 72.4° and 27.8°, how far apart are the two stadiums to the nearest meter? Answer: 2685 m Write the expression in terms of sines and/or cosines, and then simplify. 16) tan x csc x Answer: sec x 17) tan sec Answer: sin 18) 1 cos2 - 1 cot2 Answer: 1 19) tan x + 1 (sin x - 1) sec x Answer: -cos2 x 20) sec x csc x tan x cot x Answer: 1 sin x cos x Express the given trigonometric function in terms of the indicated function. 21) sin in terms of csc Answer: 22) csc 1 csc in terms of tan Answer: ± 1 + tan2 tan Use identities to find the exact value of the trigonometric function. 2 < <2 . 23) Find sin , given that cos = and 5 2 Answer: - 21 5 3 24) Find sin , given that cos Answer: = 1 and 0 < 5 < 2 . 2 6 5 25) Find tan Answer: - 26) Find sec if sec 26 and < 5 2 = <2 . 1 5 if cos = 1 and 0 < 6 < 2 . Answer: 6 27) Find sin , given that sec Answer: - = 9 77 and - < 77 2 < 0. 2 9 Use odd and even identities to simplify the expression. 28) csc x - csc (-x) Answer: 2 csc x 29) cos (-x) tan (-x) Answer: - sin x Determine whether the function is odd, even, or neither. 30) f(x) = sec(x2 ) Answer: Even 31) f(x) = x + cos x Answer: Neither 32) f(x) = x csc x Answer: Even Use identities to simplify the expression. cos2 + csc sin 33) sin2 Answer: csc2 34) 1 + sec w cos w cot2 w Answer: sec2 w 4 35) -1 sin x - csc x Answer: tan x sec x 36) cos4 x + sin2 x cos2 x Answer: cos2 x Prove that the equation is an identity. sin3 x + sin x cos2 x = cos x 37) tan x Answer: sin3 x + sin x cos2 x sin x(sin2 x + cos2 x) = tan x tan x = sin x · 1 tan x = sin x ÷ sin x cos x = sin x · cos x sin x Pythagorean identity = cos x 38) 1 1 - cos x = csc x + cot x sin x Answer: 1 1 csc x - cot x = · csc x + cot x csc x + cot x csc x - cot x = csc x - cot x csc2 x - cot2 x = csc x - cot x 1 Pythagorean identity = csc x - cot x 1 cos x = sin x sin x = 39) 1 - cos x sin x sec2 x - tan2 (-x) = sec x + tan x sec x + tan(-x) Answer: sec2 x - tan2 (-x) sec2 x - tan(-x) · tan(-x) = sec x + tan(-x) sec x + tan(-x) = sec2 x - (- tan x) · (-tan x) sec x + [- tan x] = sec2 x - tan2 x sec x - tan x = (sec x - tan x)(sec x + tan x) sec x - tan x = sec x + tan x 5 (tan(-x) = - tan x) 40) 9 csc2x - 19 csc x - 24 9 = +8 csc x - 3 sin x Answer: 9 csc2x - 19 csc x - 24 (9 csc x + 8)(csc x - 3) = csc x - 3 csc x - 3 = 9 csc x + 8 9 = +8 sin x 41) csc x cot x sin x = cot x csc x cot x Answer: csc x cot x csc x · csc x - cot x · cot x = cot x csc x cot x csc x = = 42) cos4 x = csc2x - cot2 x cot x csc x 1 cot x csc x = 1 1 · cot x csc x = 1 · sin x cot x = sin x cot x Pythagorean identity 1 - tan2 x + sin2 x tan2 x sec2 x Answer: 1 - tan2x + sin2x tan2 x = sec2 x 1- = cos2 x · 1 - sin2 x sin2 x + sin2 x · 2 cos x cos2 x 1 cos2 x sin2 x sin2 x + sin2x · cos2 x cos2x = cos2 x - sin2 x + sin4x = cos2 x - sin2 x(1 - sin2x) = cos2 x - sin2 x · cos2x = cos2 x(1 - sin2x) Pythagorean identity = cos2x · cos2 x = cos4 x Pythagorean identity 6 43) sin 1 + cos + 1 + cos sin Answer: sin 1 + cos = 2 csc + 1 + cos sin = = = 44) sin2 + 1 + 2 cos + cos2 (1 + cos )(sin ) 2(1 + cos ) = 2 csc . (1 + cos )(sin ) cos x sec x - 1 = cos x + 1 tan2x Answer: sec x - 1 = tan2x 1 -1 cos x sin2 x cos2 x cos2 x = = 45) sin2 + (1 + cos )2 (1 + cos )(sin ) 1 -1 cos x cos2x · sin2x cos2x cos x - cos2 x sin2 x = cos x(1 - cos x) 1 - cos2 x = cos x(1 - cos x) (1 - cos x)(1 + cos x) = cos x (1 + cos x) Pythagorean identity sec - 1 tan = tan sec + 1 Answer: sec - 1 (sec - 1)(sec + 1) = tan (tan )(sec + 1) = (sec2 - 1) (tan )(sec + 1) = tan2 (tan )(sec + 1) tan = . sec + 1 7 46) 1 - cos2 1 + sin Answer: 1 - = sin cos2 1 + sin = 1 + sin - cos2 1 + sin = 1 + sin + sin2 - 1 1 + sin = (sin )(1 + sin ) 1 + sin = sin . 47) (cos - sin )2 + (cos Answer: (cos + sin )2 = 2 - sin )2 + (cos + sin )2 = cos2 - 2 cos 8 sin + sin2 + cos2 + 2 cos sin + sin2 = 2.