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Practice Problems Test 2 Fall 2013 Use a calculator to find the value of the expression rounded to two decimal places. 7 15) csc-1 (- ) 3 Find the exact value of the expression. 2 ) 1) cos-1 (- 2 2) tan-1 (1) 5 16) cot-1 ( ) 8 3) sin-1 (-0.5) Simplify the trigonometric expression by following the indicated direction. 3 cos2 θ + 4 cos θ + 1 17) Factor and simplify: cos2 θ - 1 3 4) tan-1 3 Find the exact value of the expression. Do not use a calculator. π 5) cos-1 [cos ( )] 10 Complete the identity. *Note - you must practice verifying identities by completing homework problems out of the book from section 5.1 3π 6) cos-1 [cos (- )] 5 18) Use a calculator to find the value of the expression rounded to two decimal places. 6 7) cos-1 (- ) 3 19) tan2 θ - 3 sin θ tan θ sec θ = ? 20) sec 4 θ - 2 sec 2 θ tan 2 θ + tan 4 θ = ? Solve the problem. 8) Which one exists and why? sin [sin-1 (2.1)] or sin-1 [sin (2.1)] 21) 9) State the domain and range of f(x) = tan-1 x. csc θ cot θ = ? sec θ Find the exact value of the trigonometric function. 22) tan 255° Find the exact value of the expression. 3 10) cos (sin-1 ) 5 11) cos-1 (cos sec θ sin θ - 1 = ? tan θ 23) sin 7π ) 6 11π 12 Find the exact value of the expression. tan 20° + tan 10° 24) 1 - tan 20° tan 10° 7π 12) cos-1 (sin ) 6 25) 1 13) tan (cos-1 ) 3 tan 170° - tan 50° 1 + tan 170° tan 50° 26) cos 15° cos 45° - sin 15° sin 45° 14) cot-1 (-1) 1 Find the exact value under the given conditions. 4 3π 24 π 27) tan α = , π < α < ; cos β = - , < β 3 2 25 2 40) sec θ = - < π Find sin (α + β). 25 π , < θ < π 24 2 θ Find sin . 2 Use the Half-angle Formulas to find the exact value of the trigonometric function. 41) cos 22.5° Use the given information to find the exact value of the expression. 20 28) Find cos (α + β). sin α = , α lies in 29 42) sin 75° 4 quadrant I, and cos β = , β lies in quadrant I. 5 43) sin Complete the identity. π 29) cos ( + θ) = ? 2 5π 12 Solve the right triangle using the information given. Round answers to two decimal places, if necessary. 30) sin (π - θ) = ? Find the exact value of the expression. 1 3 31) sin (cos-1 - sin-1 ) 2 2 44) b = 8, α = 40°; find a, c, and β Solve the triangle. Round any answers to 2 decimal places. β = 50°, a = 3 45) α = 30°, 1 1 32) cos (sin-1 - tan-1 ) 3 2 33) sin-1 [sin ( 6π )] 7 For #ʹs 3 - 5, two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results. Round any answers to 2 decimal places. 46) a = 7, b = 9, β = 49° 34) csc-1 (-2) 2 3 35) sec-1 (- ) 3 Use the information given about the angle θ, 0 ≤ θ ≤ 2π, to find the exact value of the indicated trigonometric function. 1 Find cos (2θ). 36) cos θ = - , csc θ < 0 7 5 37) csc θ = - , tan θ > 0 2 Find cos (2θ). 4 3π 38) cos θ = , < θ < 2π 5 2 Find sin (2θ). 1 39) cos θ = , csc θ > 0 4 θ Find sin . 2 47) a = 8, b = 6, β = 15° 48) a = 5, b =69, α = 65° An object is attached to a coiled spring. The object is pulled down (negative direction from the rest position) and then released. Write an equation for the distance of the object from its rest position after t seconds. 49) amplitude = 5 cm; period = 4 seconds 50) amplitude = 11 in.; period = 8 seconds An object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in meters . Find the maximum displacement, the frequency, and the time required for one cycle. 51) d = 2 sin (5t) meters 2 52) d = -6 sin (3t) meters Solve the problem. 53) An airplane is sighted at the same time by two ground observers who are 4 miles apart and both directly west of the airplane. They report the angles of elevation as 13° and 21°. How high is the airplane? 54) A ship sailing parallel to shore sights a lighthouse at an angle of 13° from its direction of travel. After traveling 5 miles farther, the angle is 22°. At that time, how far is the ship from the lighthouse? 55) A guy wire to the top of a tower makes an angle of 53° with the level ground. At a point 32 feet farther from the base of the tower and in line with the base of the wire, the angle of elevation to the top of the tower is 25°. What is the length of the guy wire? 3 Answer Key Testname: 113REVIEWT2SP05 1) 3π 4 2) π 4 3) - π 6 4) π 6 5) π 10 6) 3π 5 7) 2.53 8) The expression sin-1 [sin (2.1)] exists because 2.1 is in the domain of the sine function, but 2.1 is not in the range of the sine function. 9) domain: all real numbers range: all real numbers 4 10) 5 11) 5π 6 12) 2π 3 13) 2 2 3π 14) 4 15) -0.44 16) 1.01 3 cos θ + 1 17) cos θ - 1 18) 0 19) -2 tan2 θ 20) 1 21) cot2 θ 3 + 2 2( 3 - 1) 4 22) 23) 24) 3 3 25) - 3 1 26) 2 27) 3 5 4 Answer Key Testname: 113REVIEWT2SP05 28) 24 145 29) -sin θ 30) sin θ 31) 0 4 10 + 5 32) 15 33) π 7 34) - 35) 5π 6 36) - 37) π 6 47 49 17 25 38) - 24 25 39) 6 4 40) 7 2 10 41) 1 2 2 + 2 42) 1 2 2 + 3 43) 1 2 2 + 3 44) a = 6.71 c = 10.44 β = 50° 45) γ = 100°, b = 4.6, c = 5.91 46) one triangle α = 35.94°, γ = 95.06°, c = 11.88 47) two triangles α1 = 20.19°, γ1 = 144.81°, c1 = 13.36 or α2 = 159.81°, γ2 = 5.19°, c2 = 2.1 48) one triangle α = 35.94°, γ = 95.06°, c = 11.88 1 49) d = -5 cos πt 2 50) d = -11 cos 1 πt 4 5 Answer Key Testname: 113REVIEWT2SP05 2 5 51) displacement = 2 meters; period = π seconds; f = oscillations/second 5 2π 3 2 oscillations/second 52) displacement = 6 meters; period = π seconds; f = 2π 3 53) 2.32 mi 54) 7.19 mi 55) 28.81 ft 6