CHAPTER 6. Section Two angles are supplementary. 2. Two angles are complementary if the sum of their measures is 90 radians

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1 SECTION 6-5 CHAPTER 6 Section 6. Two angles are complementary if the sum of their measures is 90 radians. Two angles are supplementary if the sum of their measures is 80 ( radians).. A central angle of a circle has measure radian if the length of arc opposite the angle is equal to the radius of the circle. 6. Answers will vary rotation = 5 (60 ) = rotation = 8 (60 ) = rotations = 7 6 (60 )= 0. θ = s r = 6 8 θ = 6. θ = s r = 7 8 θ = rotation= 6 () = 0. 5 rotation = 5 () = rotations= 8 () = = 0 80 = = = = 0 80 = = = 80 = = = 80 = = = = 80 = = 5 = 90 = 5 = 80. False. For example, 90 and 50 are coterminal.. False. A right angle is its own supplement and is neither acute nor obtuse. 6. False. The initial and terminal sides of k, for any integer k, coincide. 8. 8'7" = '7" =

2 5 CHAPTER 6 TRIGONOMETRIC FUNCTIONS '9" '5" = is in Quadrant II is in Quadrant III 6. is in Quadrant I 7 6. is in Quadrant IV is a quadrantal angle 68. is in Quadrant II th 70. For an angle to have degree measure, the terminal side of the angle is rotated through of a complete 60 revolution. 7. Of all the coterminal angles 0 + n60, = 80 is in the correct interval. 7. Of all the coterminal angles 80 + n60, = 80 is in the correct interval. 76. Of all the coterminal angles n, +, = and + = are in the correct interval. 78. Of all the coterminal angles + n, = and = are in the correct interval ' = C = C = (500)(60 ) 7. C = 5,000 miles 8. Linear speed = 00 ft/sec diameter = 6 ft, so C = 6 ft and 6 ft = rad. ft 00 sec rad = rad = angular speed 6 ft sec rad 60 sec sec min rev = 8. r.p.m. rad in. comes off the pulley each revolution; ft = 8 inches. s = rθ 8 = θ θ = 8 = 6 rad 9. s = r θ θ = s r = 0(5) θ = 00 = 0 radians 0 = 00 cm rear wheel front wheel Since the circumference of the earth is given by C = r, if C is known, then r can be found by using r = C/. Once the radius is known, the surface area and volume can be found using the formulas S = r and V = r /, respectively. 86. At :0 the minute hand points straight down and the hour hand has made 8 revolution angle between hands = = 90. hours = revolution 9 hours = 9 = 8 revolution 8 () = rad.6 rad 9. d = ft, so C = ft. 60 miles hr 5, 80 ft rad hr 600 sec mile ft = 88 rad sec 96. s = r θ = 8,000(0.009) = km 98. s = r θ = 500(8) 80 = ft

3 SECTION 6-55 Section 6. No. The term hypotenuse is reserved for the longest side of a right triangle.. Yes. Any altitude which falls inside the triangle will do to partition the triangle into two right triangles. 6. cos θ = a = sin (90 θ) holds for any right triangle. c 8. cot θ =. Adj = 0. cos θ = Opp 7 7 Opp = = tan θ 6. Adj Adj =. sec θ = Hyp 5 7 Opp = = sin θ 8. 5 Hyp Hyp 5 = Adj 5 Hyp = = csc θ 7 Opp 0. sin θ = θ = cos tan θ =.99 θ = sin θ = tan.99 θ.9 θ β =.7, b =. α = 90.7 = β = 6 0', c =.5 α = ' = 7 0' tan β = b a tan.7 =. a. a = tan.7 a = sin β = b c sin.7 =. c. c = = sin.7 0. α = 5, c =. β = 90 5 = 6 cos β = a c a cos 6 =. a =. cos 6 a = sin β = b c b sin 6 =. b =. sin 6 b = a =.0, b = 6. tan α = a b cos β = a c a cos 6 0' =.5 a =.5 cos 6 0' a = sin β = b c b sin 6 0' =.5 b =.5 sin 6 0' b = α = 5.7, b = 6.8 β = = 5.7 tan α = a b a tan 5.7 = 6.8 a = 6.8 tan 5.7 a = cos α = b c cos 5.7 = 6.8 c 6.8 c = = cos b = 50.0, c = 65 sin β = b c

4 56 CHAPTER 6 TRIGONOMETRIC FUNCTIONS tan α =.0 6. α = tan.0 6. α = (5 0') β = = 6.5 (6 0') c = = True. You can use the Pythagorean Theorem to find the third side, then trigonometric functions to find the angles. sin β = β = sin β = (7 0') α = = 7. (7 0') a = = True. The acute angles in a right triangle are complementary, so sin α = cos β and cos α = sin β. Thus tan α = sin α cosα = cos β sin β = cot β. True. By problem 0, sin α = cos β, so csc α = The following figures are used in Problems 5. sinα = cos β = sec β. See the figure. A line of positive slope m will form a right triangle ABC as shown. tan θ = m. Here tan θ = θ = tan = See the figure. A line of positive slope m will form a right triangle ABC as shown. tan θ = m. Here tan θ = θ = tan = 76.0 m 8. See the figure. A line of negative slope m will form a right triangle ABC as shown. tan θ =. Here tan θ = = θ = tan = See the figure. An angle θ can be formed by either of two lines, one with positive slope m and the other with negative slope m. m = ±tan θ Here m = ±tan 0 = ± See the figure. An angle θ can be formed by either of two lines, one with positive slope m and the other with negative slope m. m = ±tan θ Here m = ±tan 70 = ±.75

5 SECTION See the figure. An angle θ can be formed by either of two lines, one with positive slope m and the other with negative slope m. m = ±tan θ Here m = ±tan 0 = ± (A) sin θ = AD DC OE = AD (B) tan θ = = DC (C) csc θ = csc OED = = OE 58. (A) As θ 90, sin θ (B) As θ 90, tan θ (increases without bound) (C) As θ 90, csc θ 60. (A) As θ 0, cos θ (B) As θ 0, cot θ (increases without bound) (C) As θ 0, sec θ 6. cot α = d x h h h cot α = d x α β x = d h cot α d - x x d h cot α = h cot β d h cot β + h cot α = d h(cot β + cot α) = d d h = cot β + cot α 6. h tan. = 500 h = 500 tan. h.8668 h 5 meters cot β = x h x = h cot β 66. d = rt = 5t: 8 sin 5 0' = 5t 8 5t = sin5 0' 8 t = 5sin5 0' t hr. t min 68. ' 9,000,000 D D tan ' = 9,000,000 D = 9,000,000 tan ' D 865,708.7 D 870,000 miles to significant digits 70. The angle formed at the center of the circle is 60 = 0. Bisecting this angle to form a right 9 triangle gives 0. v 7. g = t sin θ 9.0 g =.0sin g. ft/sec

6 58 CHAPTER 6 TRIGONOMETRIC FUNCTIONS s sin 0 =.06 s =.06 sin 0 s = 8. sin inches d x 7. sec θ = tan θ = 76. Section 6 d = sec θ x = tan θ (A) Cost = C = cost along shore + cost underwater C = 0,000(0 x) + 0,000d C(θ) = 0,000(0 tan θ) + 0,000( sec θ) C(θ) = 0,000 sec θ 80,000 tan θ + 00,000 (B) θ C( θ ) = 0, 000sec θ 80, 000 tan θ + 00, $507, 75 0 $98, 58 0 $9, 76 0 $89, 5 50 $9, 7 Method : () sin 5 = () sin 5 = r = r r r + x = r + x sin5 = + r + r + x + + sin5 + sin5 r r sin5 =. inches Method : r () sin 5 = () sin 5 = r + x + r + x r sin 5 + x sin 5 = r sin 5 + r sin 5 + x sin 5 = x sin 5 = r r sin 5 x sin 5 = sin 5 r sin 5 r r sin5 sin5 r sin5 x = x = sin5 sin5 r r sin5 sin5 r sin5 = sin5 sin5 r r sin 5 = sin 5 r sin 5 r + r sin 5 = sin 5 r( + sin 5 ) = ( sin 5 ) ( sin5 ) r = inches + sin5. Answers will vary.. Since sec x = a, when a = 0 the secant is undefined. This occurs when 5 x = ±, ±, ±, Since cos x is given by a coordinate of a point on the unit circle, cos x, so y = cos x lies between y = and y =.

7 SECTION W() = (, 0) 0. W 5 = (0, ). W =,. W =, 0. W ( ) = (, 0), so csc = 0 is not defined W 6 =, 5 W = (0, ), so cos 5 = W = ( ) 0,. W =,, so cot = =. 6. w =,, so tan = =. 8. w 5 6 =,, so sec 5 6 = =. 0. w 9 = ( ) 0, so sin 9, =. tan x > 0 for W(x) in quadrants I or III. sec x > 0 for W(x) in quadrants I or IV 6. csc x < 0 for W(x) in quadrants III or IV 8. sin sec(.555) 6.. cot tan csc cos 5 '7" True 5. True. A similar statement could be made about any function. 5. False, but only false because cot x and tan x are undefined for an infinite set of real numbers. If both cot x and tan x are defined then the statement holds. 56. True. A similar statement could be made about any function. 58. True. Both are defined for all real numbers. 60. Graph y = sin x using trig axes on [0, ]. The zeros are 0,,,,, and the turning points are,,,, 5,, and 7,. cos x 6. Graph y = or y = tan x sin x using trig axes on [0, ]. The zeros are 5 7,,, and. There are no turning points. 6. W() is in quadrant I; a, +; b, W() is in quadrant III; a, ; b, 68. W(7) is in quadrant I; a, +; b, W(.5) is in quadrant II; a, ; b, + 7. W(.8) is in quadrant III; a, ; b, 7. W(x) = (, 0) x =, x = + k, k any integer 76. W(x) =, x = 7 7, x = + k, k any integer 78. W(x) is the coordinates of a point on a unit circle that is x units from (, 0), in a counterclockwise direction if x is positive and in a clockwise direction if x is negative. W(x 6) has the same coordinates as W(x), since we return to the same point every time we go around the unit circle any integer multiple of units (the circumference of the circle) in either direction.

8 60 CHAPTER 6 TRIGONOMETRIC FUNCTIONS 80. cos x > 0: QI, QIV tan x < 0: QII, QIV both: QIV 8. sin x > 0 and csc x < 0 is never true because sin x = they have same sign. csc x 8. sin x is defined for all x, 0 x 86. cot x is not defined for x = 0,, 88. csc x is not defined for x = 0,, 90. b(ka) a(kb) = akb akb = 0 9. A = nr sin = n ()() sin 8 in 9. A = nr sin = n (8)(0) sin = 00 8 = 00 = cm 96. a = a = + cos.500 a = cos a = cos a 5 = Section sin x and (sinx) have the same meaning; the square of the sine of x. sinx means the sine of the square of x.. Draw a perpendicular from a point P = (a, b) on the terminal side of the angle to the horizontal axis. The resulting right triangle is the reference triangle. 6. If a periodic function has one zero, so that f(a) = 0, then f(a + p) = 0, f(a + p) = 0, and so on. It has therefore infinitely many zeros (for example, sin x). Otherwise it can have no zeros (for example, sec x) 8. function period cosine tangent secant 0. (A) The graph of y = sin x deviates unit from the x axis. (B) The graph of y = cot x deviates indefinitely far from the x axis. (C) The graph of y = sec x deviates indefinitely far from the x axis.. (A) y = cos x: [, ] (B) y = tan x: [, ] (C) y = sec x: [, ] x intercepts:,,, x intercepts:,, 0,, x intercepts: none. (A) y = sin x is defined for all x on [, ] (B) y = cot x is not defined for x =,, 0,, on [, ] (C) y = sec x is not defined for x =,,, on [, ] 6. vertical asymptotes: (A) y = sin x: none (B) y = cot x:,, 0,, (C) y = sec x:,,,

9 SECTION (A) Shifting y = sec x, to the right produces y = csc x. 0. (B) Shifting y = sec x, to left and a reflection in the x axis produces y = csc x. y = sec x + cos( x) cos x sec( x) sec x = = = = y x x x x y is odd.. cos( x) cos x cos x sin( x) sin x sin x cot( x) cot x = = = = x x x x x y is even.. x sin( x) cos( x) = x( sin x)(cos x) = x sin x cos x = y y is even. 8. x + sin ( x) = x + [sin( x)] = x + ( sin x) = x + sin x y or y y is neither even nor odd.. (, ); r = ( ) + = 5; QII sin θ = 5, cos θ = 5, tan θ = csc θ = 5, sec θ = 5, cot θ = 6. x sin( x) = x + sin x = y y is odd. 0. ( x) sin( x) = x ( sin x) = x sin x = y y is even.. (, ); r = ( ) + = ; QI sin θ =, cos θ =, tan θ = csc θ =, sec θ =, cot θ = 6. α = 5 8. α = 0. α =. sin θ = θ = 0 = rad. tan θ = 6. sec θ = θ = 0 = rad θ = 5 = rad 8. tan θ = (QII, QIV) and sin θ < 0 (QIII, QIV) QIV a =, b = : r = a + b r = + ( ) r = 5 tan θ = b a = sin θ = b r = 5 cos θ = a r = 5 cot θ = a b = csc θ = r b = 5 sec θ = r a = cos θ = 5 (QII, QIII) and tan θ > 0 (QI, QIII) QIII a = 5, r = : a + b = r ( 5 ) + b = b = b =

10 6 CHAPTER 6 TRIGONOMETRIC FUNCTIONS cos θ = a r = 5 sec θ = r a = 5 sin θ = b r = csc θ = r b = tan θ = b a = 5 cot θ = a b = 5 = = When the terminal side of an angle lies along the horizontal axis, y = 0 which implies csc θ = r b and cot θ = a b are not defined. 5. cot θ =, 0 θ < 60 Ref = 60, QII, QIV: θ = 0, True. If f(x) = 0, f( x) = 0 and f( x) = 0 = 0. f 60. True. If f(x + p) = f(x) and g(x + p) = g(x) then (x + p) = g 6. False. In fact if both f and g are even, so is fg. 56. sec θ =, 0 θ < Ref =, QII, III: θ = 5, f ( x + p) = g( x + p) f ( x) f = (x) for all x. g( x) g 6. If f(x + p) = f(x) for all x, then a(x + p) + b(x + p) + c = ax + bx + c a(x + p ) + bp = 0 For non-zero p this can only be true for all x if a = b = 0. Hence f(x) = c, c any real number. 66. If f( x) = f(x) for all x, then a( x) + b = (ax + b) ax + b = ax b b = 0 b = 0 Hence f(x) = ax, a any real number. 68. (A) (B) The x intercepts do not change. (C) The graphs deviated,, and units from the X axis. (D) The amount of deviation from the x axis appears to be A. 70.(A) (B),, and (C) For y = cos nx, n a positive integer, n periods would appear in this viewing window. 7.(A) (B) The graph of y = sin(x + C) is shifted C units to the left for C > 0 and C units to the right for C < 0.

11 SECTION In all three cases the calculator gives a division by zero error. (A) csc = sin = (B) tan = (C) cot 0 = h(x) = tan x, g(x) = x - For x close to zero the two graphs are almost identical. 78. (A) s = r θ 8 = θ θ = radians a a (B) cos θ = cos = a.07 b b sin θ = sin = b.5 (.07,.5) tan θ = r = = θ = + ( ) = 6 = s = r θ = = 8. I = k cos θ = n A = sin n (A) n = 8: A = 8 n = 00: A = 00 n =,000: A = 000 k cos k k cos k k cos 90 = k(0) = 0 8. y = sin 6t + t, t 0 6 (cos 6 ) y(0.) = sin(6(0.)) + 6 (cos(6 (0.))) = sin.88 (B) A =.596 as n 8.95 sin n = 0,000: A = 0, m = tan θ, 0 θ 80 sin sin.59 0, 000 (A) θ = 5. : m = tan θ = 9.0 : m = tan y ( ) =.9(x 6) y + =.9x y =.9x Section 6 5. Answers will vary. (B) θ = 06 : m = tan 06 =.9 through (6, ):. No. Amplitude is defined for functions of the form y = A sin(bx + C) and y = A cos(bx + C) and some others, but not for all trigonometric functions. 6. Amplitude = A = = Period = = = B 8. Amplitude = A = = Period = = = B

12 6 CHAPTER 6 TRIGONOMETRIC FUNCTIONS 0. Amplitude is not defined for the tangent function. Period = B =. Amplitude is not defined for the cotangent function. Period = = B =. Amplitude is not defined for the secant function. Period = = B = 6. Amplitude A = =. Period = = B =. The basic cosine function cos t has zeros when t = + k, k an integer. We examine x = + k x, x = when x =,,,. + k, and find that x falls in the interval x 8. Amplitude is not defined for the tangent function. Period = =. The basic tangent function tan t has zeros when t = k, k an integer. We examine x = k, < x <, and find x = k falls in this interval when x = 0 and x =. 0. Amplitude = A = =. Period = = =. The basic sine function sin t has turning points B k when t = + k, k an integer. We examine x = + k, x, and find that x = + falls in this 8 interval when x = ,,,,,,,. The turning points are therefore, , 5, 8,, 8, 8,, 8,,, 8, 5, 8, 7, 8.. Amplitude is not defined for the cosecant function. Period = = =. B The basic cosecant function csc t has turning points when t = + k, k an integer. We examine x = + k, 0 < x < 8, and find that x = + k falls in this interval when x =,, 5, 7. The turning points are therefore (, ), (, ), (5, ), (7, ).. From the graph, A =, P = 8 = 8 B = B y = sin x, x 8 8. From the graph, A = 0., P = = B = 8 B y = 0. cos 8x, 8 x 6. From the graph, A =, reflected about the x axis, P = = B = B y = sin x, x 0. From the graph, A =, reflected about the x axis, P = 0.5 = B B = y = cos(x), 0.5 x 0.5

13 SECTION y = 5 sin x Amplitude: 5; period: ; phase shift: 0. Amplitude: ; period: ; phase shift: 6. Period: ; phase shift: 8. Period: ; phase shift: 0 0. Amplitude: ; period: 8; phase shift: 0. y = cos[(x )] Amplitude = A = =. Period: = ; phase shift: Note that the graph is an upside-down cosine curve.

14 66 CHAPTER 6 TRIGONOMETRIC FUNCTIONS. Period: ; phase shift: 6. y = 8 sec x; period: = ; phase shift: 0 8. False. The graph of y = cos x does not pass through the origin. 50. True. If f(x) = A cos Bx, then f( x) = A cos B( x) = A cos Bx = f(x). 5. False. The function y = sin x has period y = cot x + tan x, P = 60. y = csc x cot x, P = x y = csc x y = tan

15 SECTION y = sec x 6. y = cot x 66. From the graph, Amplitude = and Period = = B = B Phase Shift = C B = B = C = y = sin x +, x 68. From graph, Amplitude = reflected across the x axis, and Period = 8 = 8 B = B Phase Shift = C B = C = C = y = cos x +, x y = 5. sin ( ).5 t, 0 t 6 Amplitude = 5. Period = Phase Shift =.5.5 = = y = 5 cos[5(t 0.)], 0 t Amplitude = 5 Period = 5 = 5 Phase Shift = 0.5 = 0. 5

16 68 CHAPTER 6 TRIGONOMETRIC FUNCTIONS 7. y = sin x 76. y = sin x y = 5 sin(x +.88) 80. The amplitude is decreasing. The amplitude is decreasing with time. This is often referred to as a damped sine wave. Examples are a car's vertical motion, which is damped by the suspension system after the car goes over a bump, and the slowing down of a pendulum that is released away from the vertical line of suspension (air resistance and friction). 8. The amplitude is increasing The amplitude is increasing with time. In physical and electrical systems this is referred to as resonance. Some examples are the swinging of a bridge during high winds and the movement of tall buildings during an earthquake. Some bridges and buildings are destroyed when the resonance reaches the elastic limits of the structure I = 0 sin 0t A = 0 P = = 0 60 frequency = 60 Hz

17 SECTION E = A cos Bt A = 0 P = 60 = B = 0 B E = 0 cos(0t), t 0 t 90. V(t) = cos, 0 t 8 9. I = 0 cos(0t ), 0 t 60 A = 0, P = 0 = 60, PS = 0 = 0 The graph shows the volume of air in the lungs t seconds after exhaling. 9. θ = + 6t y = sin θ = sin6 t + y = sin 6 t +, 0 t a t 96. (A) tan θ = a = 0 tan 0 (B) (C) Initially the distance from N is zero. The distance increases slowly at first, then begins to increase rapidly without end. 98. (A) (Max y Min y) 76 (B) A = = =.5 B = B = 6 k = Min y + A = +.5 = 5.5 C. x y = sin. 6

18 70 CHAPTER 6 TRIGONOMETRIC FUNCTIONS (C) 8 (D) 0 0 Section 6 6. On this interval, the function is not one-to-one since, for example cos = cos but.. No. For example tan (tan ) = 0. tan (tan x) = x only for function. < x <, the restricted domain of the tangent 6. Yes. f(a) < f(b) whenever a < b implies (for a one-to-one function) a < b whenever f(a) < f(b), thus f (a) < f (b) whenever a < b. 8. sin 0 = 0 0. arccos = 6. cos = 6. arctan = 6. tan = 8. tan 0 = 0 0. cos tan arcsin. is undefined. 6. arccos ( ) is undefined. 8. sin = 0. arctan( ) =. sec[sin ( 0.099)] tan tan[ tan ( 0)] = 0 8. sin[sin 5 ] is not defined. 0. cos[cos ( )] =.. sin sin = sin ( ) =. For the identity sin (sin x) = x to hold, x must be in the restricted domain of the sine function; that is, x. The number is not in the restricted domain. cos cos = 5. For the identity cos (cos x) = x to hold, x must be in the restricted domain of the 5 cosine function; that is, 0 x. The number is not in the restricted domain tan tan = tan =. For the identity tan (tan x) = x to hold, x must be in the restricted domain of the tangent function; that is, < x <. The number 5 is not in the restricted domain. 8. cos ( ) = sin ( ) is not defined. 5. arctan( ) = tan (.0) arccos( 0.906) 57.0

19 SECTION cos [cos( 0.5)] = 0.5. For the identity cos (cos x) = x to hold, x must be in the restricted domain of the cosine function; that is, 0 x. The number 0.5 is not in the restricted domain. 60. True. The restricted trigonometric functions are one-to-one, hence so are their inverses. 6. True. sin x, cos x, tan x <, cot x <, sec x, csc x. 6. False. cos = ; cos = (A) - (B) The graph is the same. The domain of the inverse sine is the interval [, ] for < x < 0, < cos x < - sin(cos x) = x

20 7 CHAPTER 6 TRIGONOMETRIC FUNCTIONS for 0 x <, 0 < cos x sin(cos x) = x sin(cos x) = x in both cases. 78. for < x 0 < sin x 0 tan(sin x) = x x for 0 < x <, 0 < sin x < tan(sin x) = x x tan(sin x x) = in both cases. x 80. Solve for x: y = 5 sin(x ) y = sin(x ) 5 sin y = x f (x) = + sin x sin y = x 5 x for 5 5 x 5 x 8 8. (A) - - (B) The domain for sin x is (, ) and the range is [, ], which is the domain for sin x.

21 SECTION Thus, y = sin (sin x) has a graph over the interval (, ), but sin (sin x) = x only on the restricted domain of sin x, [ /, /] mm: θ = tan.6 x 70 mm: θ = tan.6 x - = tan.6 7 = tan (A) (B) 0 0 focal length 7.8 mm L = D + (d D) θ + C sin θ = D + (d D)cos D d C + C sin D d cos C = (6) + ( 6)cos (0)sincos (0) (0) = 6 cos (0.) + 0 sin(cos (0.)) = 5.8 inches 90. (A) 60 0 (B) Distance between centers 0.0 in d = r tan x r = (50)tan 5 50 = 00 tan feet

22 7 CHAPTER 6 TRIGONOMETRIC FUNCTIONS Chapter 6 Group Activity (Summarizing and Discussion are left to the student) (A).. Compute the regression equation. Graph the data and the regression equation. (B).. Compute the regression equation. Graph the data and the regression equation.