Math 104 Final Exam Review
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1 Math 04 Final Exam Review. Find all six trigonometric functions of θ if (, 7) is on the terminal side of θ.. Find cosθ and sinθ if the terminal side of θ lies along the line y = x in quadrant IV.. Find the remaining trigonometric functions of θ if cscθ = and θ terminates in quadrant II. 4. Use identity substitutions to simplify: a) cscθ cotθ cosθ b) ( cos θ)(+ cos θ) 5. Show that cos θ(secθ + tan θ) = + sinθ by transforming the left side into the right side. 6. Simplify the expression 9 x as much as possible after substituting sinθ for x. 7. If sinθ = with θ in quadrant I, find cosθ, cscθ, and cotθ. a 8. In which quadrant will θ lie if cscθ > 0 and cosθ < 0? 9. In ABC, C = 90, c= 4.79 cm, and b =.68 cm. Draw the triangle then find each of the following: a) Side a b) Angle A c) Angle B 0. In ABC C = 90, A = 60, and side a = cm. Find exact answers for each of the following. a) Side c b) Side b. Use a calculator to find: a) tan 6 50 your answers to two decimal places) b) θ if θ is acute and secθ =.9 (Round. A CB antenna is located on the top of a garage that is 6 feet tall. From a point on level ground that is 00 feet from a point directly below the antenna, the antenna subtends an angle of. Approximate the length of the antenna to one decimal place. 6 00
2 . A pilot, flying at an altitude of 5000 feet, wishes to approach a landing point on a runway at an angle of 0 (angle of depression). Approximate, to the nearest 00 feet, the distance from the airplane to the landing point at the beginning of the descent. 4. Give the exact values of: a) sin b) 4cos 4 c) 5 csc 6 5. Find the exact values of: a) sec 45 b) csc 60 c) sin 60 + cos 45 (Simplify) 6. Show that cotangent is an odd function. 7. Convert to radians: a) 0 8. Convert to degrees: a) 4 b) 50 b) 7 9. Draw the following angles in standard position and find the reference angle: a) 7 b) 0 c) 0. Use a calculator to find θ if θ is between 0 and 60 and a) cos θ =.477 with θ in quadrant III (Round to the nearest tenth.) b) secθ =.545 with θ in quadrant IV (Round to the nearest tenth.). If θ = is a central angle that subtends an arc length of circle. s =, find the radius of the 4. Find the area of the sector formed by central angle θ =.4 in a circle of radius r = cm.. A conical paper cup is constructed by removing a sector from a circle of radius 5 inches and attaching edge OA to OB (see figure). Find angle AOB so that the cup has a depth of 4 inches. B O A A B O 4
3 4. Find the amplitude, period, and phase shift of: y = cos x 4 y = 5sin x Sketch one period of: y = sin x y = cos x+ 6 c. y = 6cot ( x) d. y = 6csc x 6. Find the equations of each of the following:
4 c. 7. Find an interval over which the graph of y = cos x+ completes one complete cycle. 8. Find the range of y = cos x+ (+ ) Find the exact value for each of the following: sin sin sin sin 4 c. sin sin 0. Find the exact values for each of the following: 9 cos cos( ) + 9 cos cos( )
5 . Find the exact value for each of the following: sin tan 4 cos sin 6 c. sin arccos d. 4 sin arctan arccos 5. If x, rewrite cos(sin x) is terms of x without trig functions. 4. Let sinθ = with θ in the second quadrant and 5 Compute sin( θ + α). sinα = with α in the third quadrant. 4. Derive the formula for tan( A B) using the formulas for sine and cosine. Find tan A if tan( A B) = 9 and tan B =. 5. If x is a positive number, find: cos(sin x) tan(cos x) 6. Find the exact values of: sin.5 tan c. cot5 7. Let sin A = with A in quadrant IV and 5 sin B = with B in quadrant II and find: A sin( A+ B) cos B c. sin
6 8. Prove the following identities: cos x + sin x = sin x cos x sec x= sin x(tan x+ cot x) c. sec x cos x= sin xtan x d. sin x cot x = cos x 9. If x= asinθ, θ and a > 0, express a x in terms of a trig function of θ. 40. Express as a single trig function and then simplify: 5 5 sin cos + cos sin 4. Solve sinθ = 0 for 0 θ Solve θ + θ = for 0 θ 60 cos 5cos 0 4. Solve cos x+ sin x = 0 for 0 x 44. Solve cscθ + cotθ = 0 for 0 θ Solve sinθ = cosθ for 0 θ 46. Find all radian solutions for sin 4θ cos 4θ = 47. Solve triangle ABC given B = 57, C =, and side a = 7. meters. 48. A man standing near a radio station antenna observes that the angle of elevation to the top of the antenna is A. He then walks a distance s further away and observes that the angle of elevation to the top of the antenna is B. Find the height of the antenna to the nearest foot. Assume A = 6, B = 44 and s = 0 feet.
7 49. How many triangles ABC satisfy the following conditions? A = 40, b = 87 ft., and a = 6 ft. 50. In triangle ABC, A = 7, b = 48cm, and a = 9 cm. Find angle B. 5. In triangle ABC, if a = 0 m, b = 0 m, and c = 40 m, find the measure of the smallest angle. 5. Two planes leave an airport at the same time. Their speeds are 60 mph and 40 mph and the angle between their courses is 8. How far apart are they after.5 hours? 5. Express the polar equation r = 6sin θ in rectangular form. 54. Write the pair (, ) in polar coordinates. 55. Sketch the graph of r = + 4cosθ
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