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Pre-Calculus 6.0 FINAL EXAM Review Packet Name: __________________ Date: ___________ Pd: ___ Graph the given angle and then find a positive and negative coterminal angle. = θ 173° 1.) 2.) pos: _____ neg: _____ θ= − 7π 6 pos: _____ neg: _____ θ= 3.) 19π 3 pos: _____ neg: _____ Convert to Decimal Degree. Round to 4 decimal places. 4.) −145°54'37 " = _________ 5.) 23°41'46" = ___________ Convert to DMS. Round to the nearest second. 6.) 241.7614° = __________ 7.) 1.36 = ___________ Convert from degrees to radian measure without a calculator. 8.) 105° = _______ 9.) 135° = ________ 10.) 50° = ________ Convert from radians to degrees without using a calculator. 11.) 7π = _______ 4 12.) − 17π 13π = _________ 13.) = ________ 3 6 Find the length of the arc (s) intercepted by a central angle ( θ ) of a circle with given dimension. Round to 2 decimal places and include units of measure. 14.) θ = π 7 and diameter = 16” 15.) = θ 122.6° and radius = 2.1 m. Pre-Calculus 6.0 FINAL EXAM Review Packet Find the central angle of a circle, in DMS rounded to the nearest second, for the following situations: 16.) The arc length intercepted by a central angle is 3”. The diameter of the circle is 1.7” 17.) The arc length intercepted by a central angle is equal to the radius of the circle. Solve for linear speed and/or angular speed as requested. 18) A car is moving at 35 mph. What is the linear speed of a pebble that is stuck in the tread of the 20” diameter tires on the car in feet per second? What is the rotational speed of the tires, in rpm? What is the angular speed of the tires under these conditions? 19.) The rotational speed of a pulley on a drill press is 2500 rpm. If the pulley is 4” in diameter, find the angular speed of the pulley and the linear speed of a drive belt which rides on the edge of the pulley. Find the values of the six trig functions: 20.) sin θ = _____ cscθ = _____ cosθ = _____ secθ = _____ tan θ = _____ cotθ = _____ θ 8 3 2 Pre-Calculus 6.0 FINAL EXAM Review Packet Set up a trig equation, then solve for the indicated values. Round to 2 decimal places as necessary. 21.) 22.) 13 x 7 4 x θ 41° y θ = _________ (DMS) x = _________ X = ________ y = ________ 23.) 24.) θ 11 7 2π 5 8 x X = ________ θ = __________ (Dec Radian) Use your calculator to find the indicated values. Round to 4 decimal places. 25.) sin 123° = ______ 26.) cos 54°12’ = _______ 27.) cot 54°34” = ______ 28.) csc 85.51° = _______ 29.) sec 25.73° = _______ 30.) tan 54.22° = _______ 31.) sec q = -2.1503; q = _______ 32.) cot q = 0.1766; q = ________ 33.) cos q = 1.0449; q = _______ 34.) tan q = 2.1266; q = ________ 35.) csc q = 0.4819; q = _______ 36.) csc q = 1.4819; q = _______ Pre-Calculus 6.0 FINAL EXAM Review Packet 2 and sec θ < 0 , find the value of the other 5 5 trigonometric functions, expressed in simplest radical form. Given sin = − 37.) sin θ = _____ cscθ = _____ cosθ = _____ secθ = _____ tan θ = _____ cotθ = _____ Find all angles over [0, 2) that meet the given criteria. Express answer in radian measure. If answer is not a common angle, round to 4 decimal places. 38.) sin θ = 3 11 40.) csc θ = − 7 3 39.) cos θ = − 2 7 41.) cot θ = − 4 9 42.) A six foot Eskimo is standing 20 ft from the base of a heptangonal prism ice castle. His eyes are about 6” below the top of his head. He can see the top of the castle if he looks up at an angle of 63 . Find the height of the ice castle. 43.) A shepherd is cautiously watching over his flock of sheep from the top of a 75 foot cliff. Assuming that the shepherd is lying on the ground at the top of the cliff to avoid being spotted by predators, at what angle of depression does the shepherd need to look in order to keep an eye on a rare orange spotted sheep that is located 92 feet from the base of the cliff? Express answer in DMS format. Pre-Calculus 6.0 FINAL EXAM Review Packet 44.) The top of a 200 foot lighthouse is spotted at an angle of elevation of 27from a ship that is headed straight towards the lighthouse. About 15 minutes later, lighthouse keeper who lives at the top of the lighthouse, observes the ship at an angle of depression of 33 . How far did the ship move and what was its speed over the last 15 minutes? Express answer in feet and feet per min respectively. 45.) From a certain location on a street leading straight to a tall building, the angle of elevation to the top of the building is 24 . From another location on the same street that is 500 feet away from the first location, the angle of elevation is measured to be 30 . Find the height of the building. Building A B Ground Find the reference angle for the following: 46.) 210° 47.) 300° 48.) -97° 49.) 384° 50.) -205° 51.) 53.) 3.3571 54.) 4.1950 52.) − 7π 11 5π 3 In which quadrant(s) do the following conditions hold true? 55.) sin q > 0 and sec q < 0 ______ 56.) sec q > 0 and cot q < 0 ______ Pre-Calculus 6.0 FINAL EXAM Review Packet Find the complement and supplement of the following angles. Answer in the same units of measure as given in the problem. 3π 5π 57.) 58.) 59.) 123 7 9 Identify the amplitude, period, phase shift and vertical shift for each of the following. 60.) y = 5 cos (2q - p) 61.) y = 5 − Amp: __________ Period: __________ Phase Shift: ________ Vertical Shift: _________ 1 sin q 2 Amp: __________ Period: __________ Phase Shift: ________ Vertical Shift: _________ π 1 62.) y = 1 - sin θ + 2 2 Amp: __________ Period: __________ Phase Shift: ________ Vertical Shift: _________ 3π 63.) y = -3 cos 3x + 2 Amp: __________ Period: __________ Phase Shift: ________ Vertical Shift: _________ Graph the following: 4 3 64.) y = cos (q + p) 2 1 - 2π - 3π 2 -π -π 2 -1 -2 -3 -4 π 2 π 3π 2 2π 5π 2 3π 7π 2 4π Pre-Calculus 6.0 FINAL EXAM Review Packet π 1 65.) y = 2 sin x − 2 2 4 3 2 1 -π 2 -π - 3π 2 - 2π π 2 -1 π 3π 2 5π 2 2π 4π 7π 2 3π -2 4 66.) y = sec (x - p) -3 3 -4 2 1 - 2π Asymptotes: x = - 3π 2 -π -π 2 π 2 -1 π 3π 2 5π 2 2π 4π 7π 2 3π -2 -3 -4 4 67.) y = 2 csc x 3 2 1 - 2π Asymptotes: x = - 3π 2 -π 2 -π π 2 -1 π 3π 2 2π 5π 2 3π 7π 2 -2 -3 -4 4 68.) y = tan x 3 2 1 - 2π Asymptotes: x = - 3π 2 -π -π 2 π 2 -1 π 3π 2 5π 2 2π 4π 7π 2 3π -2 -3 -4 4 69.) y = cot x 3 2 1 - 2π Asymptotes: x = - 3π 2 -π -π 2 -1 -2 -3 -4 π 2 π 3π 2 2π 5π 2 3π 7π 2 4π 4π Pre-Calculus 6.0 FINAL EXAM Review Packet Evaluate the following. Answers must be in simplest rational, radical form. 3 70.) arcsin − 71.) sin −1 ( −1 ) = ___________ = ________ 2 2 72.) arccos 73.) tan −1 3 = ___________ = _________ 2 3 74.) sin arcsin = ________ 75.) tan ( arctan ( −549 ) ) = _____ 8 1 76.) cos ( arccos π ) = ________ 77.) tan arcsin − = ______ 2 ( ) ( 78.) cos arc cot 3 79.) cos arctan − = _____ 3 2 81.) tan arcsin − = _____ 5 4π 83.) arcsin sin = _____ 3 ( 3 )) = _____ 3 80.) sin arc cos = _____ 7 5π 82.) arc cos cos = _____ 4 Complete the chart with the domain and range of the following functions using interval notation where possible: Function 84) 85) 86) 87) 88) 89) 90) 91) 92) sin x cos x tan x csc x sec x cot x arcsin x arccos x arctan x Domain Range Graph the following: 93) y=arcsinx 94) y=arccosx 95) y=arctanx Pre-Calculus 6.0 FINAL EXAM Review Packet Simplify using the trig identities: 96.) − tan( −θ ) = ________ π cos − θ 2 97.) sin2 x − 1 = _________ csc2 x − 1 Find the general solutions to the given equations as well as the specific solutions over [0, 2Π). 98.) 2 cos x + 1 = 0 99.) 2 sin2 x = 1 100.) 3tan3 x = tan x 101.) 2 sin x − 102.) sin(2x ) − 4 cos x = 0 103.) 1 cos x = 0 3 1 x −1 sin − 4 cos(x ) = 2 2 State the case (SSS, SAS, AAS or SSA), whether or not it is ambiguous, and then use Law of Sines or the Law of Cosines to solve if possible. DRAWINGS ARE NOT TO SCALE. A 104.) 80° 9.5 70° B C Pre-Calculus 6.0 FINAL EXAM Review Packet 105.) A 15 14 60° B C A 106.) 7 63° B C 23 A 107.) 14 8 B C 10 Find the area of the following triangles: A 108.) A 109.) 7 34° B B C 5 Area = _________ 14 6 8 Area = __________ C A 111.) 5 B 17 Area = ___________ A 110.) 12 7 C B 35° 8 Area = ____________ C