Advanced Math (2) Semester 1 Review Name SOLUTIONS

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1 dvanced Math (2) Semester Review Name SOLUTIONS IMPORTNT!!!!! PLESE EMIL ME IF YOU FIND TYPO!. Find the value of x to the nearest yard. 53 YRDS x 5 yards 32 cos32 = 5 x 5 x = cos32 x = 53 yards 2. In D, =, = 65, and a =. Find side b to the nearest tenth. b=5.2 sin = sin 65 b b = sin65 sin = In D, = 57, a = 8, and b = 7. Find the number of solutions to the triangle. ONE sin57 8 = sin 7 = sin - 7sin 57 = 7.2 or so only solution 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 (Q and Q don t have same sign for csc (or sin).)

2 6. Find the equation of the graph below. y = 2sinx ϖ 2ϖ 7. Find the equation of the graph below. y=2cos[(2ϖ/3)(x )] y= 2cos[(2ϖ/3)(x 2.5)] y= 2sin[(2ϖ/3)(x 0.25)] y= 2sin[(2ϖ/3)(x.75)] 3ϖ 8. Graph the equation y = -2sin p (x - 3) Simplify. a) csc coscot b) - coscos - cos 2 sin 2 sec csc tan csc Á Ë c) sin 2 cot tan csc 2 sin2 cos 2 sin 2 tan sin2 sin2 ˆ 2

3 0. What quadrant is x in if cos x < 0 and sin x < 0? Q3. Evaluate csc p 6. since sin p 2 = 2, csc p 6 =2 2. Graph y = 2 - cos x p ˆ Á. Ë 3 - p 3 5p 3. onvert p 5 to degrees. 5. Find the length of side to the nearest tenth x 5 sin 60 = sin 5 0 = or.7 = = 8.7 x 2 = cos8.7 x 2 = x = What is the horizontal shift of the graph of y = -3cos x - p ˆ Á? nswer: p to the right. Ë Find the equation that matches the graph below: y = 3 cos 2x ϖ 2ϖ 3ϖ 3

4 8. Find cosq if q is an angle in standard position with its terminal ray passing through the point ( 2, ). cos x = = ( 2, ) q 9. cot = cot = adj opp = 2 2 = = 8 = If sec x = 2 find all values of x to the nearest degree for 0 x < 360. x = 60, If sec x is positive, in what quadrants could x be? Quadrants I or IV. 22. If tanx = 3 find all values of x to the nearest degree for 0 x < 360. x = 60, Graph a full period of the following function. y = 2sinÁ p Ë 2 x ˆ One period goes from x=0 to x=.

5 2. 50 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? q = cosq = - 5 and sin q > 0. Find tan q. tan q = Find the exact value of cos 75. cos 75 = Find the exact value of sin 2 cos 2 - cos 2 sin 2. sin (2 2 ) = sin 30 = 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 (aa) formula. cos 2a = cos (aa) = cos a cos a sin a sin a = cos 2 a sin 2 a 30. Solve the following equations for LL values of x in radians in the interval [0, 2ϖ). Ï a tanx sinx sin x = 0 x = Ì 0, 3p, p, 7p Ó SET = 0, FTOR sinx ( tanx ) = 0 sinx = 0 or tanx = 0 b) sin x cos 2x - cos x sin 2x = - Ï 7p x = Ì 2 2, ϖ Ó 2, 9ϖ 2, 23ϖ 2 This simplifies using the sin(a b) formula: ( ) = - 2 sin x - 2x spin TWIE sin2x = - 2 2x = 7p 6,p 6,9p 6, 23p 6 x = 7p 2,p 2,9p 2, 23p 2 5

6 c) sin 2 x - = 0 sin 2 x = / sin x = ±/2 PLUS OR MINUS! Ï p x = 6, 5ϖ 6, 7ϖ 6, ϖ Ì Ó 6 d) 3 sin x = sin x = /3 NO SOLUTION! 3. Find LL possible angles and sides. = 86.5, a = 8.6, = = 3.5, a 2 =., 2 = REMEMER sin - x has solutions in QI and QII Find the missing angles. = 23.2, = 0.3, = Solve each equation on the interval 0 x < 360. a) 3 csc x - 5 = 0 csc x = 5/3 sin x = 3/5 x = {36.9, 3. } b) 2 sin 2 x - sin x - = 0 (2 sin x )( sin x ) = 0 sin x =.5, sin x = x = {90, 20, 330 } c) tan 2 x - = 0 (tan x ) ( tan x ) = 0 tan x = or tan x = x= {5, 35, 225, 35 } 6

7 3. Verify the identity: 35. Verify the identity: cosq sinq sinq cosq = 2secq cosq( - sinq) sinq ( sinq)(- sinq) cosq = cosq(- sinq) - sin 2 q cosq(- sinq) cos 2 q - sinq sinq = cosq 2 cosq = 2secq = sinq cosq = sinq cosq = cot 2 q sinq cscq = cscq cscq cot 2 q cscq sinq sinq = csc 2 q - cscq = (cscq -)(cscq ) = cscq cscq = 36. 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? 2.7 miles x port x 2 = (250)(75)(cos75 ) x = 2.7 b) What course must the boat sail to return to port? N 62. E sin 75 = =7. course is

8 37. Label the points whose coordinates are: a) (, -20 ) b) (-5, 35 ) c) (3, 300 ) d) D(-, -50 ) D 38. onvert to rectangular coordinates. a) (5, -225 ) b) (-, 50 ) c) (3, 270 ) ,5 2 ˆ Á 2 3,-2 Ë 2 ( ) (0,-3) 39. onvert to polar coordinates. Round to the nearest degree. 7 3 a) (-, -) b) ( 2, ) c) (-7, 0) ( 7,9 ) ( 7,330 ) ( 7,80 ) 0. Find four polar coordinates for the point (2, 0 ). a) (2, ) b) (2, ) c) ( 2, ) d) ( 2, ) (2, 00) b) (2, 320 ) c) ( 2, 220 ) d) ( 2, 0 ). Find the equation of the following graph. y = sec x 8

9 2. Read the following carefully, then fill in the blank. a) If tan q = k, then tan ( q) = k. b) If sec q = k, then sec ( q) = k. c) If cos q = r, then cos (q 2ϖ)= _r. d) If tan q= r, then tan (qϖ) = r. 3. Find the smaller angle made between the two hands of a clock at : =66.5. 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 2p 9=7.5ϖ Planet Moog is 2.8 x 0 7 km from Earth and has an apparent size of about What is the approximate diameter of planet Moog? You don't need to evaluate your expression for this diameter p = 88.7 km 6. onvert the following. a) 2.2 to degrees, minutes, and seconds = = = b) decimal degrees. (I changed this problem!) = 76. 9