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Advanced Math (1 ) IMPORTANT!!!!! Semester Exam Review #2 SOLUTIONS Name _______________ PLEASE LET YOUR TEACHER KNOW IF YOU FIND A TYPO! 1. Find the value of x to the nearest yard. 53 YARDS ˙ 45 cos32° = x x 45 x= cos32° 32° x = 53 yards 45 yards 2. In !ABC, "A = 44° , "B = 65° , and a = 4 . Find side b to the nearest tenth. b=5.2 sin44° sin 65° = 4 b 4sin65° b= = 5.2 sin 44° 4 44° 65° B A 3. In !ABC, "A = 57° , a = 8, and b = 7 . Find the number of solutions to the triangle. ONE sin57° sin B = 8 7 7sin 57° B = sin !1 = 47.2° or 132.8° 8 138.2° + 57° !180° so only 1 solution. 8 7 57° B A 4. The angle of depression from the top of a cliff 750 m high to the base of a streetlight is 68 ° . To the nearest meter how far is the street light from the foot of the cliff? 303 feet 5. Does csc x = csc (-x) for all values of x? NO (Q1 and Q4 don’t have same sign for csc (or sin).) 1 6. Find the equation of the graph below. y = 2sinx + 1 π 3π 2π 7. Find the equation of the graph below. y=2cos[(2π/3)(x–1)]–1 # 8. Graph the equation y = "2sin (x " 3) + 1. 2 ! 9. Expand and simplify: = sin 2x 1! cos 2x 2 sin x cos x ( 1! cos2 x ! sin2 x ) 2 sin x cos x 1! cos2 x + sin2 x 2 sin x cos x = sin2 x + sin2 x 2 sin x cos x = 2 sin2 x cos x = sin x = cot x = 2 10. What quadrant is x in if cos x < 0 and sin x < 0? Q3 ! 11. Evaluate csc . 2 6 # "& 12. Graph y = 2 ! cos% x + ( . $ 3' " # 3 5" 3 14. Convert 4! to degrees. 144° 5 ! ! 15. Find the length of side BC to the nearest tenth. 12.7 or 2.27 A 14 15 B 14 2 = 15 2 + x 2 " 2 #15 # x # cos60° 60° 15 0 = x 2 "15x + 29 x = 12.7, or x = 2.27 C x ! # "& ! 16. What is the horizontal shift of the graph of y = !3cos% x ! ( ? Answer: to the right $ 2' 2 17. Find the equation that matches the graph below: y = –3 cos 2x π 3π 2π 3 18. Find cos! if θ is an angle in standard position with its terminal ray passing through the !2 5 =! point (–2, 4). cos x = 5 2 5 (–2, 4) θ 19. cot C = _____ cot C = A adj 2 21 21 = = opp 4 2 10 4 C B 2 2 10 ! 4 = 84 = 2 21 20. If sec x = 2 find all values of x to the nearest degree for 0° ! x < 360° . x = 60°, 300° 21. If sec x is positive, in what quadrants could x be? Quadrants I or IV. 22. If tan x = 3 find all values of x to the nearest degree for 0° ! x < 360° . x = 60°, 240° "! % 23. Graph a full period of the following function. y = 2sin$ x ' #2 & 2 -2 4 24. A 150 ft building casts a shadow of 250 feet at 3:00 pm. What is the angle of elevation from the end of the shadow to the top of the building? θ = 31° 4 3 and sin θ > 0. Find tan θ. tan θ = ! 5 4 6! 2 26. Find the exact value of cos 75 ° . cos 75° = 4 25. cos! = " 27. Find the exact value of sin 42 ° cos 12 ° - cos 42 ° sin 12 ° . sin (42° – 12°) = sin 30° = 0.5 28. Derive the formula for sin 2a from the sin (a + a) formula. sin 2a = sin (a + a) = sin a cos a + cos a sin a = 2 sin a cos a 29. Derive the formula for cos 2a from the cos (a+a) formula. cos 2a = cos (a+a) = cos a cos a – sin a sin a = cos2 a – sin2 a 30. Solve the following equations for ALL values of x in radians in the interval 0 ! x < 2" a b) # 3" 7" & tanx sinx + sin x = 0 x = $0, , ", ' SET = 0, FACTOR 4 4( % sinx ( tanx + 1) = 0 sinx = 0 or tanx + 1 = 0 ! " 7! 11! 19! 23! % 1 sin 4x cos 2x - cos 4x sin 2x = ! x=# , , , & spin TWICE 2 $ 12 12 12 12 ' This simplifies using the sin(a–b) formula: 1 sin(4x ! 2x) = ! 2 1 sin2x = ! 2 7" 11" 19" 23" 2x = , , , 6 6 6 6 7" 11" 19" 23" x= , , , 12 12 12 12 5 c) 4 sin2 x - 1 = 0 d) 3 sin x = 4 sin2 x = 1/4 sin x = ±1/2 PLUS OR MINUS! % ! 5! 7! 11! " x=& , , , # 6 6 $ '6 6 sin x = 4/3 NO SOLUTION! 31. Find ALL possible angles and sides. A 12 A1 = 86.5°, a1 = 18.6, C1 = 53.5° A2 = 13.5°, a2 = 4.4, C2 = 126.5° REMEMBER sin-1x has solutions in QI and QII. 15 C 40° B 32. Find the missing angles. A 7 C 3 B A = 123.2°, B = 40.3°, C = 16.5° 9 ° ° 33. Solve each equation on the interval 0 ! x < 360. a) 3 csc x - 5 = 0 2 b) 2 sin x - sin x - 1 = 0 2 c) tan x - 1 = 0 csc x = 5/3 sin x = 3/5 x = {36.9°, 143.1°} (2 sin x + 1 )( sin x – 1) = 0 sin x = –.5, sin x = 1 x = {90°, 210°, 330°} (tan x + 1) ( tan x – 1) = 0 tan x = –1 or tan x = 1 x= {45°, 135°, 225°, 315°} 6 34. Verify the identity: cos " 1+ sin " + = 2sec " 1+ sin " cos" cos" (1# sin " ) ! 35. Verify the identity: ! 1+ sin " = (1+ sin " )(1# sin " ) cos " cos" (1# sin " ) 1+ sin " + = cos" 1# sin 2 " cos" (1# sin " ) 1+ sin " + = cos" cos2 " 1# sin " + 1+ sin " = cos " 2 = cos" 2sec " = + cot 2 " + sin " csc " = csc " 1+ csc " cot 2 ! +1= 1+ csc !! cot 2 ! +1+ csc ! = 1+ csc ! csc 2 ! + csc ! = 1+ csc ! csc ! csc ! +1 = 1+ csc ! csc ! = csc ! ! ( ) 36. A boat sails from port on a course N30 ° W for 75 miles and then turns to a SW course for 250 miles. a) How far must the boat sail to return to port? 241.7 miles b) What course must the boat sail to return to port? 7 N 62.44° E 37. Label the points whose coordinates are: a) A(4, -240°) b) B(-5, 135°) c) C (3, 300°) d) D(-1, -150°) A D C B 38. Convert to rectangular coordinates. a) (5, -225°) b) (-4, 150°) " 5 2 5 2% $! , ' 2 & # 2 (2 c) (3, 270°) 3,!2 ) (0,!3) 39. Convert to polar coordinates. Round to the nearest degree. 7 3 7 ! a) (-4, -1) b) ( 2 , 2 ) c) ( 17,194°) (-7, 0) (7,330°) (7,180°) 40. Find four polar coordinates for the point (2, –40°). a) 41. (2, – ) b) (2, + ) (2, – 400) b) (2, + 320°) c) (–2 , – c) (–2 , – 220°) d) Find the equation of the following graph. y = sec x 8 ) d) (–2 , + ) (–2 , +140°) 42. Read the following carefully, then fill in the blank. a) If tan θ = k, then tan (–θ) = ____–k _______ b) If sec θ = k, then sec (–θ) = _____ k _______ c) If cos θ = r, then cos (θ + 2π)= ___ r _____. d) If tan θ = r, then tan (θ + π)= ____ r ____. 43. Find the smaller angle made between the two hands of a clock at 4:57. 120°+6°⋅3+ 57 " 30° =166.5° 60 ! 44. The minute hand of a clock is 9 inches long. How far (in inches) has the tip of the minute hand moved (around in a circle) in 25 minutes? 25 " 2# " 9 =7.5π 60 45. ! Planet Moog is 2.8 x 107 km from Earth and has an apparent size of about 0.001°. What is the approximate diameter of planet Moog? You don't need to evaluate your expression for this diameter. .001 " 2# " 2.8 "10 7 = 488.7 km 360 46. a) Convert the following. ! 42.21° to degrees, minutes, and seconds. 42° 0.21⋅60’ = 42° 12.6’ = 42° + 12’ + 0.6⋅60’’=42° 12’ 36’’ b) 76°6’36’’ to decimal degrees. 6 36 76+ + = 76.11 60 3600 ! 9