15 hij 60 _ip = 45 = m 4. 2 _ip 1 huo 9 `a = 36m `a/_ip. v 41

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1 Name KEY Math 2 Final Review Unit 7 Trigonometric Functions. A water wheel has a radius of 8 feet. The wheel is rotating at 5 revolutions per minutes. Find the linear speed, in feet per second, of the water. 5 hij k`a 2m 8 no hij k`a 9m no/_ip 60 _ip 2. A large pizza from Papa John s has a radius of 9. a) It is cut into eight equal pieces. What angle, in degrees and convert to radians, will be formed by a slice? m b) Before the pizza was made, the crust was thrown into the air and spun a complete rotation every ½ second. What was the speed of the outer edge of the crust in inches per second? huo 2m 2 _ip huo 9 `a 36m 3.0 `a/_ip 3. A tower that is 98 feet tall casts a shadow of 3 feet long. Find the angle of elevation of the sun to the nearest degree. tan v wx ccd v. The angle of elevation to the top of a pole is 62. If the distance to the pole is 7 feet, what is the height of the pole? tan 62 y ze { 88. niio 5. Find the exact value of cos(_`a bc d ) without using your calculator. Show your diagram for e full credit. (LT 8C) 3 7 v

2 6. Find the exact value of tan( cos z ). (LT 8C) ~ v 7. Find the exact value of each of the remaining trigonometric functions of v if sin v, sec v > 0. ~ sin v ~ csc v ~ v cos v c ~ tan v b c c c sec v ~ ~ c c c cot v c b 8. Solve for x in the equation on the interval [0, 2m). Solve algebraically and give exact values for the answers. a) tan { + 0 tan { dñ and eñ z z b) sin { + c 0 sin { 2 eñ Ö ccñ and Ö 9. Solve cos(x) + 2 cos(x) sin x 0 on the interval [0, 2π). + 2 sin { 0 ( + 2 sin {) 0 0 and + 2 sin { 0 Ñ and dñ 2 sin { sin { 2 eñ Ö and ccñ Ö

3 0. Using standard notation for a triangle, solve triangle ABC if A 30, a, and b 52. sin 30 sin ã 52 ã 36 à sin sin 30 p p 80.. Using standard notation solve the triangle if á 30, à 0, p. sin 0 sin 30 å å 7.6 ã 0 sin 0 sin 0 ç ç.8 2. Two ships leave a harbor at the same time. Ship A travels on a bearing of 9 at 0 mph to location A. The other ship, Ship B, travels on a bearing of 65 at 2 mph to location B. How far apart, to the nearest mile, will the ships be after three hours? 65 2 mph for 3 hrs. is 36 miles mph for 3 hrs. is 30 miles { (36)(30) cos 26 { 59 miles 3. A surveyor needs to find the distance between two houses that are separated by a lake. She measures angle ACB, which is found to be 5, and then walks off the distance to each house, 80 feet and 00 feet, respectively. How far apart are the houses? á { 00 { (00)(80) cos 5 { 3 feet 5 ã 80 à

4 Unit 8 Graphs of Trigonometric Functions. Find the amplitude, period, phase shift, and vertical shift of è 6 sin 8{ m + 2. Graph the function. Amplitude: 6 Period: Ñ Ñ x z Phase Shift: Ñ x Vertical Shift: π/ -3π/6 -π/8 -π/6-2 π/6 π/8 3π/6 π/ y x 5. Find the amplitude, period, phase shift, and vertical shift of è 2 cos(3({ Ñ )) 2. Graph the d function. Amplitude: 2 Period: Ñ d Phase Shift: Ñ d Vertical Shift: π/3 -π/2 -π/3 -π/6 - π/6 π/3 π/2 2π/ y x

5 Unit 9 Analytic Trigonometry 6. Simplify cos({) sec( {). 7. Simplify êëê y íìî ï y ëñê y ëóî ï yòëóî y. csc { tan { sec { sin { + sin { sin { cos { sin { + sin { sec { õ sin { ú { ùsin cos { û sin { + sin { sin { cos { sin { + sin { sin { 8. Prove or disprove the following identity: (sec(x) tan(x)) cbëóî(ö) còëóî(ö). sin { cos { sin { + sin { cos { sin { + sin { sin { (sin { + sin { ) sin { sin { (sin { + ) (sin { + ) (sec { tan {) ù sin { û ù sin { û ( sin {)( sin {) ( sin {) ( sin {)( + sin {) + sin { ( sin {)( sin {) cos { 9. Prove _`a {pu_ { + pu_ z { pu_ {. sin { cos { + cos z { cos { (sin { + cos {) cos { (2) cos { 20. Prove sin { + cot { csc {. sin { + cot { sin { + õ sin { ú sin { + ( sin { ) sin { sin { + sin { sin { + cos { sin { sin { csc {

6 2. Find all of the solutions of 2sin x cos x 0. 2 sin { 0 2( cos {) cos { 0 2 cos { + 0 (2 cos { + ) 0 (2 cos { + 2 ) 0 2 ( + ) ( + ) 0 ( + )(2 ) 0 ( + )(2 ) c m + 2ma Ñ d + 2ma and ~Ñ d + 2ma 22. Prove cos 2ü cos ü sin ü cos 2ü cos(ü + ü) cos ü cos ü sin ü sin ü cos ü sin ü 23. Find the exact value of each of the following under the given conditions: cos x 2 25, 3π 2 < x < π ; tan(y) 6, π 2 < y < π 25 7 { a) sin(x y) 2 sin({ è) sin { cos è sin è ù 7 25 û ù 2 û ù 7 25 û 6 7 ß è 6 b) tan({ è) tan({ è) (tan { tan è) + tan { tan è õ 7 2 ú 6 ß + õ 7 2 ú 6 ß ß ( ) 2. Find the exact value of the expression below: Give that c, for { in Quadrant IV and tan è d for è in Quadrant I find cos({ è). cd z 3 2 { 5 cos({ è) cos è + sin { sin è 5 è 3 ù 2 3 û ù 5 û + ù 5 3 û ù3 5 û

7 Unit 0 Vector, Parametric, and Polar Coordinates 25. Let áã be determined by the points á(2, 3) and ã(, ). What is the component form of áã? 2, 3 3, Let be a vector from initial point c to terminal point. Write in terms of the and vectors. c ( 8, 5) and ( 3, ) 3 8, 5 5, 5` 27. Let 6 2, and Æ 3 8. Find Æ. ü ( 3` 8 ) (6` 2 ) 3` 8 6` + 2 9` 6 Æ ( 9) + (6) Let 2 + 3, 6 3, and Æ. Find 3 (2Æ ). 3( µ + ) (2( ) ( ) 6` + 9 ( 8` 6` + 3 ) 6` ` + 6` 3 8` Consider force vectors ü and j acting on the same point. Find the resultant magnitude and angle v Ø. ü 200 pounds, v 29 j 300 pounds, v cos 29, 200 sin cos 205, 300 sin , Magnitude: ( 96.97) + ( 29.82) 0.5 Angle: tan v w.x bwö.we v 7

8 30. The magnitude and direction of two forces acting on an object are 60 pounds, S 0 E, and 70 pounds N 62 E, respectively. Find the magnitude, to the nearest hundredth of a pound, and the direction angle, to the nearest tenth of a degree, of the resultant force cos 28, 70 sin cos( 50 ), 60 sin( 50 ) 00.37, 3.0 Magnitude: (00.37) + ( 3.0) 0.22 pounds Angle: tan v bcd.c c.de v 7. so ª 83 º 3. Find the horizontal and vertical components of the vector with magnitude 0 and direction v 20. Write the vector in terms of the vectors and. 0 cos 20, 0 sin , ` Find the measure of the angle between the vectors h 3, and _ 6, 8. Round your answer to the nearest degree. Explain why the vectors are orthogonal or not orthogonal. ( 3)(6) + ()(8) cos v 25 (0) cos v 50 v 7 Not orthogonal because the angle between the vectors is not 90 degrees. 33. A force of 60 pounds on a rope is used to pull a box up a ramp inclined at 20 from the horizontal. The rope forms an angle of 3 with the horizontal. How much work is done pulling the box 22 feet along the ramp? 60 cos 23, 60 sin 23 22, 0 60 cos 23, 60 sin 23,25.07 nuuo Ωuüaæ_

9 3. Given the point h, v (2, dñ z ): a) Plot the point in the polar coordinate system. b) Give another representation of the point in polar form using a negative r value. ( 2, 7m ) c) Convert the point to rectangular coordinates (x, y). (Leave as exact values) { 2 cos 3m ß è 2 sin 3m ß ( 2, 2) 35. Given the point h, v ( 3, eñ c ): a) Plot the point in the polar coordinate system. b) Give another representation of the point in polar form using a negative r value. (3, 9m 2 ) c) Convert the point to rectangular coordinates (x, y). { 3 cos 7m è 3 sin 7m (0.78, 2.90)