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13.2.6: Chapter 6

  • Page ID
    117281
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    Try It

    6.1 Graphs of the Sine and Cosine Functions

    1.

    6π 6π

    2.

    1 2 1 2 compressed

    3.

    π 2 ; π 2 ; right

    4.

    2 units up

    5.

    midline: y=0; y=0; amplitude: | A |= 1 2 ; | A |= 1 2 ; period: P= 2π | B | =6π; P= 2π | B | =6π; phase shift: C B =π C B =π

    6.

    f( x )=sin(x)+2 f( x )=sin(x)+2

    7.

    two possibilities: y=4sin( π 5 x π 5 )+4 y=4sin( π 5 x π 5 )+4 or y=4sin( π 5 x+ 4π 5 )+4 y=4sin( π 5 x+ 4π 5 )+4

    8.
    A graph of -0.8cos(2x). Graph has range of [-0.8, 0.8], period of pi, amplitude of 0.8, and is reflected about the x-axis compared to it's parent function cos(x).

    midline: y=0; y=0; amplitude: | A |=0.8; | A |=0.8; period: P= 2π | B | =π; P= 2π | B | =π; phase shift: C B =0 C B =0 or none

    9.
    A graph of -2cos((pi/3)x+(pi/6)). Graph has amplitude of 2, period of 6, and has a phase shift of 0.5 to the left.

    midline: y=0; y=0; amplitude: | A |=2; | A |=2; period: P= 2π | B | =6; P= 2π | B | =6; phase shift: C B = 1 2 C B = 1 2

    10.

    7

    A graph of 7cos(x). Graph has amplitude of 7, period of 2pi, and range of [-7,7].
    11.

    y=3cos( x )4 y=3cos( x )4

    A cosine graph with range [-1,-7]. Period is 2 pi. Local maximums at (0,-1), (2pi,-1), and (4pi, -1). Local minimums at (pi,-7) and (3pi, -7).

    6.2 Graphs of the Other Trigonometric Functions

    1.
    A graph of two periods of a modified tangent function, with asymptotes at x=-3 and x=3.
    2.

    It would be reflected across the line y=1, y=1, becoming an increasing function.

    3.

    g(x)=4tan(2x) g(x)=4tan(2x)

    4.

    This is a vertical reflection of the preceding graph because A A is negative.

    A graph of one period of a modified secant function, which looks like an downward facing prarbola and a upward facing parabola.
    5.
    A graph of one period of a modified secant function. There are two vertical asymptotes, one at approximately x=-pi/20 and one approximately at 3pi/16.
    6.
    A graph of one period of a modified secant function, which looks like an downward facing prarbola and a upward facing parabola.
    7.
    A graph of two periods of both a secant and consine function. Grpah shows that cosine function has local maximums where secant function has local minimums and vice versa.

    6.3 Inverse Trigonometric Functions

    1.

    arccos(0.8776)0.5 arccos(0.8776)0.5

    2.
    1. π 2 ; π 2 ;
    2. π 4 ; π 4 ;
    3. π; π;
    4. π 3 π 3
    3.

    1.9823 or 113.578°

    4.

    sin −1 (0.6)=36.87°=0.6435 sin −1 (0.6)=36.87°=0.6435 radians

    5.

    π 8 ; 2π 9 π 8 ; 2π 9

    6.

    3π 4 3π 4

    7.

    12 13 12 13

    8.

    4 2 9 4 2 9

    9.

    4x 16 x 2 +1 4x 16 x 2 +1

    6.1 Section Exercises

    1.

    The sine and cosine functions have the property that f( x+P )=f( x ) f( x+P )=f( x ) for a certain P. P. This means that the function values repeat for every P P units on the x-axis.

    3.

    The absolute value of the constant A A (amplitude) increases the total range and the constant D D (vertical shift) shifts the graph vertically.

    5.

    At the point where the terminal side of t t intersects the unit circle, you can determine that the sint sint equals the y-coordinate of the point.

    7.
    A graph of (2/3)cos(x). Graph has amplitude of 2/3, period of 2pi, and range of [-2/3, 2/3].

    amplitude: 2 3 ; 2 3 ; period: 2π; 2π; midline: y=0; y=0; maximum: y= 2 3 y= 2 3 occurs at x=0; x=0; minimum: y= 2 3 y= 2 3 occurs at x=π; x=π; for one period, the graph starts at 0 and ends at 2π 2π

    9.
    A graph of 4sin(x). Graph has amplitude of 4, period of 2pi, and range of [-4, 4].

    amplitude: 4; period: 2π; 2π; midline: y=0; y=0; maximum y=4 y=4 occurs at x= π 2 ; x= π 2 ; minimum: y=4 y=4 occurs at x= 3π 2 ; x= 3π 2 ; one full period occurs from x=0 x=0 to x=2π x=2π

    11.
    A graph of cos(2x). Graph has amplitude of 1, period of pi, and range of [-1,1].

    amplitude: 1; period: π; π; midline: y=0; y=0; maximum: y=1 y=1 occurs at x=π; x=π; minimum: y=1 y=1 occurs at x= π 2 ; x= π 2 ; one full period is graphed from x=0 x=0 to x=π x=π

    13.
    A graph of 4cos(pi*x). Grpah has amplitude of 4, period of 2, and range of [-4, 4].

    amplitude: 4; period: 2; midline: y=0; y=0; maximum: y=4 y=4 occurs at x=0; x=0; minimum: y=4 y=4 occurs at x=1 x=1

    15.
    A graph of 3sin(8(x+4))+5. Graph has amplitude of 3, range of [2, 8], and period of pi/4.

    amplitude: 3; period: π 4 ; π 4 ; midline: y=5; y=5; maximum: y=8 y=8 occurs at x=0.12; x=0.12; minimum: y=2 y=2 occurs at x=0.516; x=0.516; horizontal shift: 4; 4; vertical translation 5; one period occurs from x=0 x=0 to x= π 4 x= π 4

    17.
    A graph of 5sin(5x+20)-2. Graph has an amplitude of 5, period of 2pi/5, and range of [-7,3].

    amplitude: 5; period: 2π 5 ; 2π 5 ; midline: y=−2; y=−2; maximum: y=3 y=3 occurs at x=0.08; x=0.08; minimum: y=−7 y=−7 occurs at x=0.71; x=0.71; phase shift: −4; −4; vertical translation: −2; −2; one full period can be graphed on x=0 x=0 to x= 2π 5 x= 2π 5

    19.
    A graph of -cos(t+pi/3)+1. Graph has amplitude of 1, period of 2pi, and range of [0,2]. Phase shifted pi/3 to the left.

    amplitude: 1 ; period: 2π; 2π; midline: y=1; y=1; maximum: y=2 y=2 occurs at x=2.09; x=2.09; maximum: y=2 y=2 occurs at t=2.09; t=2.09; minimum: y=0 y=0 occurs at t=5.24; t=5.24; phase shift: π 3 ; π 3 ; vertical translation: 1; one full period is from t=0 t=0 to t=2π t=2π

    21.
    A graph of -sin((1/2)*t + 5pi/3). Graph has amplitude of 1, range of [-1,1], period of 4pi, and a phase shift of -10pi/3.

    amplitude: 1; period: 4π; 4π; midline: y=0; y=0; maximum: y=1 y=1 occurs at t=11.52; t=11.52; minimum: y=1 y=1 occurs at t=5.24; t=5.24; phase shift: 10π 3 ; 10π 3 ; vertical shift: 0

    23.

    amplitude: 2; midline: y=3; y=3; period: 4; equation: f(x)=2sin( π 2 x )3 f(x)=2sin( π 2 x )3

    25.

    amplitude: 2; period: 5; midline: y=3; y=3; equation: f(x)=2cos( 2π 5 x )+3 f(x)=2cos( 2π 5 x )+3

    27.

    amplitude: 4; period: 2; midline: y=0; y=0; equation: f(x)=4cos( π( x π 2 ) ) f(x)=4cos( π( x π 2 ) )

    29.

    amplitude: 2; period: 2; midline y=1; y=1; equation: f( x )=2cos( πx )+1 f( x )=2cos( πx )+1

    31.

    0,π 0,π

    33.

    sin(π2)=1 sin(π2)=1

    35.

    π2 π2

    37.

    f(x)=sinx f(x)=sinx is symmetric

    39.

    π3,5π3 π3,5π3

    41.

    Maximum: 1 1 at x= 0 x=0 ; minimum: -1 -1 at x= π x=π

    43.

    A linear function is added to a periodic sine function. The graph does not have an amplitude because as the linear function increases without bound the combined function h(x)=x+sinx h(x)=x+sinx will increase without bound as well. The graph is bounded between the graphs of y=x+1 y=x+1 and y=x-1 y=x-1 because sine oscillates between −1 and 1.

    ed12b8b4f8ed195e3c71a4adbc3dc8143afcc161
    45.

    There is no amplitude because the function is not bounded.

    7cdb1cbd174f5d300e3bccc17d07531ea9e0f4ab
    47.

    The graph is symmetric with respect to the y-axis and there is no amplitude because the function’s bounds decrease as |x| |x| grows. There appears to be a horizontal asymptote at y=0 y=0 .

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    6.2 Section Exercises

    1.

    Since y=cscx y=cscx is the reciprocal function of y=sinx, y=sinx, you can plot the reciprocal of the coordinates on the graph of y=sinx y=sinx to obtain the y-coordinates of y=cscx. y=cscx. The x-intercepts of the graph y=sinx y=sinx are the vertical asymptotes for the graph of y=cscx. y=cscx.

    3.

    Answers will vary. Using the unit circle, one can show that tan( x+π )=tanx. tan( x+π )=tanx.

    5.

    The period is the same: 2π. 2π.

    7.

    IV

    9.

    III

    11.

    period: 8; horizontal shift: 1 unit to left

    13.

    1.5

    15.

    5

    17.

    cotxcosxsinx cotxcosxsinx

    19.
    A graph of two periods of a modified tangent function. There are two vertical asymptotes.

    stretching factor: 2; period: π 4 ; π 4 ; asymptotes: x= 1 4 ( π 2 +πk )+8, where k is an integer x= 1 4 ( π 2 +πk )+8, where k is an integer

    21.
    A graph of two periods of a modified cosecant function. Vertical Asymptotes at x= -6, -3, 0, 3, and 6.

    stretching factor: 6; period: 6; asymptotes: x=3k, where k is an integer x=3k, where k is an integer

    23.
    A graph of two periods of a modified tangent function. Vertical asymptotes at multiples of pi.

    stretching factor: 1; period: π; π; asymptotes: x=πk, where k is an integer x=πk, where k is an integer

    25.
    A graph of two periods of a modified tangent function. Three vertical asymptiotes shown.

    Stretching factor: 1; period: π; π; asymptotes: x= π 4 +πk, where k is an integer x= π 4 +πk, where k is an integer

    27.
    A graph of two periods of a modified cosecant function. Vertical asymptotes at multiples of pi.

    stretching factor: 2; period: 2π; 2π; asymptotes: x=πk, where k is an integer x=πk, where k is an integer

    29.
    A graph of two periods of a modified secant function. Vertical asymptotes at x=-pi/2, -pi/6, pi/6, and pi/2.

    stretching factor: 4; period: 2π 3 ; 2π 3 ; asymptotes: x= π 6 k, where k is an odd integer x= π 6 k, where k is an odd integer

    31.
    A graph of two periods of a modified secant function. There are four vertical asymptotes all pi/5 apart.

    stretching factor: 7; period: 2π 5 ; 2π 5 ; asymptotes: x= π 10 k, where k is an odd integer x= π 10 k, where k is an odd integer

    33.
    A graph of two periods of a modified cosecant function. Three vertical asymptotes, each pi apart.

    stretching factor: 2; period: 2π; 2π; asymptotes: x= π 4 +πk, where k is an integer x= π 4 +πk, where k is an integer

    35.
    A graph of a modified cosecant function. Four vertical asymptotes.

    stretching factor: 7 5 ; 7 5 ; period: 2π; 2π; asymptotes: x= π 4 +πk, where k is an integer x= π 4 +πk, where k is an integer

    37.

    y=tan( 3( x π 4 ) )+2 y=tan( 3( x π 4 ) )+2

    A graph of two periods of a modified tangent function. Vertical asymptotes at x=-pi/4 and pi/12.
    39.

    f( x )=csc( 2x ) f( x )=csc( 2x )

    41.

    f( x )=csc( 4x ) f( x )=csc( 4x )

    43.

    f( x )=2cscx f( x )=2cscx

    45.

    f(x)= 1 2 tan(100πx) f(x)= 1 2 tan(100πx)

    47.
    A graph of the absolute value of the cotangent function. Range is 0 to infinity.
    49.
    A graph of tangent of x.
    51.
    A graph of two periods of a modified secant function. Vertical asymptotes at multiples of 500pi.
    53.
    A graph of y=1.
    55.
    1. ( π 2 , π 2 ); ( π 2 , π 2 );
    2. A graph of a half period of a secant function. Vertical asymptotes at x=-pi/2 and pi/2.
    3. x= π 2 x= π 2 and x= π 2 ; x= π 2 ; the distance grows without bound as | x | | x | approaches π 2 π 2 —i.e., at right angles to the line representing due north, the boat would be so far away, the fisherman could not see it;
    4. 3; when x= π 3 , x= π 3 , the boat is 3 km away;
    5. 1.73; when x= π 6 , x= π 6 , the boat is about 1.73 km away;
    6. 1.5 km; when x=0 x=0
    57.
    1. h( x )=2tan( π 120 x ); h( x )=2tan( π 120 x );
    2. An exponentially increasing function with a vertical asymptote at x=60.
    3. h( 0 )=0: h( 0 )=0: after 0 seconds, the rocket is 0 mi above the ground; h( 30 )=2: h( 30 )=2: after 30 seconds, the rockets is 2 mi high;
    4. As x x approaches 60 seconds, the values of h( x ) h( x ) grow increasingly large. The distance to the rocket is growing so large that the camera can no longer track it.

    6.3 Section Exercises

    1.

    The function y=sinx y=sinx is one-to-one on [ π 2 , π 2 ]; [ π 2 , π 2 ]; thus, this interval is the range of the inverse function of y=sinx, y=sinx, f(x)= sin 1 x. f(x)= sin 1 x. The function y=cosx y=cosx is one-to-one on [ 0,π ]; [ 0,π ]; thus, this interval is the range of the inverse function of y=cosx,f(x)= cos 1 x. y=cosx,f(x)= cos 1 x.

    3.

    π 6 π 6 is the radian measure of an angle between π 2 π 2 and π 2 π 2 whose sine is 0.5.

    5.

    In order for any function to have an inverse, the function must be one-to-one and must pass the horizontal line test. The regular sine function is not one-to-one unless its domain is restricted in some way. Mathematicians have agreed to restrict the sine function to the interval [ π 2 , π 2 ] [ π 2 , π 2 ] so that it is one-to-one and possesses an inverse.

    7.

    True . The angle, θ 1 θ 1 that equals arccos(x) arccos(x), x>0 x>0, will be a second quadrant angle with reference angle, θ 2 θ 2 , where θ 2 θ 2 equals arccosx arccosx, x>0 x>0. Since θ 2 θ 2 is the reference angle for θ 1 θ 1 , θ 2 =π θ 1 θ 2 =π θ 1 and arccos(x) arccos(x) = πarccosx πarccosx-

    9.

    π 6 π 6

    11.

    3π 4 3π 4

    13.

    π 3 π 3

    15.

    π 3 π 3

    17.

    1.98

    19.

    0.93

    21.

    1.41

    23.

    0.56 radians

    25.

    0

    27.

    0.71

    29.

    -0.71

    31.

    π 4 π 4

    33.

    0.8

    35.

    5 13 5 13

    37.

    x1 x 2 +2x x1 x 2 +2x

    39.

    x 2 1 x x 2 1 x

    41.

    x+0.5 x 2 x+ 3 4 x+0.5 x 2 x+ 3 4

    43.

    2x+1 x+1 2x+1 x+1

    45.

    2x+1 x 2x+1 x

    47.

    t t

    49.
    A graph of the function arc cosine of x over -1 to 1. The range of the function is 0 to pi.

    domain [ 1,1 ]; [ 1,1 ]; range [ 0,π ] [ 0,π ]

    51.

    approximately x=0.00 x=0.00

    53.

    0.395 radians

    55.

    1.11 radians

    57.

    1.25 radians

    59.

    0.405 radians

    61.

    No. The angle the ladder makes with the horizontal is 60 degrees.

    Review Exercises

    1.

    amplitude: 3; period: 2π; 2π; midline: y=3; y=3; no asymptotes

    A graph of two periods of a function with a cosine parent function. The graph has a range of [0,6] graphed over -2pi to 2pi. Maximums as -pi and pi.
    3.

    amplitude: 3; period: 2π; 2π; midline: y=0; y=0; no asymptotes

    A graph of four periods of a function with a cosine parent function. Graphed from -4pi to 4pi. Range is [-3,3].
    5.

    amplitude: 3; period: 2π; 2π; midline: y=4; y=4; no asymptotes

    A graph of two periods of a sinusoidal function. Range is [-7,-1]. Maximums at -5pi/4 and 3pi/4.
    7.

    amplitude: 6; period: 2π 3 ; 2π 3 ; midline: y=1; y=1; no asymptotes

    A sinusoidal graph over two periods. Range is [-7,5], amplitude is 6, and period is 2pi/3.
    9.

    stretching factor: none; period: π; π; midline: y=4; y=4; asymptotes: x= π 2 +πk, x= π 2 +πk, where k k is an integer

    A graph of a tangent function over two periods. Graphed from -pi to pi, with asymptotes at -pi/2 and pi/2.
    11.

    stretching factor: 3; period: π 4 ; π 4 ; midline: y=2; y=2; asymptotes: x= π 8 + π 4 k, x= π 8 + π 4 k, where k k is an integer

    A graph of a tangent function over two periods. Asymptotes at -pi/8 and pi/8. Period of pi/4. Midline at y=-2.
    13.

    amplitude: none; period: 2π; 2π; no phase shift; asymptotes: x= π 2 k, x= π 2 k, where k k is an odd integer

    A graph of two periods of a secant function. Period of 2 pi, graphed from -2pi to 2pi. Asymptotes at -3pi/2, -pi/2, pi/2, and 3pi/2.
    15.

    amplitude: none; period: 2π 5 ; 2π 5 ; no phase shift; asymptotes: x= π 5 k, x= π 5 k, where k k is an integer

    A graph of a cosecant functionover two and a half periods. Graphed from -pi to pi, period of 2pi/5.
    17.

    amplitude: none; period: 4π; 4π; no phase shift; asymptotes: x=2πk, x=2πk, where k k is an integer

    A graph of two periods of a cosecant function. Graphed from -4pi to 4pi. Asymptotes at multiples of 2pi. Period of 4pi.
    19.

    largest: 20,000; smallest: 4,000

    21.

    amplitude: 8,000; period: 10; phase shift: 0

    23.

    In 2007, the predicted population is 4,413. In 2010, the population will be 11,924.

    25.

    5 in.

    27.

    10 seconds

    29.

    π 6 π 6

    31.

    π 4 π 4

    33.

    π 3 π 3

    35.

    No solution

    37.

    12 5 12 5

    39.

    The graphs are not symmetrical with respect to the line y=x. y=x. They are symmetrical with respect to the y y -axis.

    A graph of cosine of x and secant of x. Cosine of x has maximums where secant has minimums and vice versa. Asymptotes at x=-3pi/2, -pi/2, pi/2, and 3pi/2.
    41.

    The graphs appear to be identical.

    Two graphs of two identical functions on the interval [-1 to 1]. Both graphs appear sinusoidal.

    Practice Test

    1.

    amplitude: 0.5; period: 2π; 2π; midline y=0 y=0

    A graph of two periods of a sinusoidal function, graphed over -2pi to 2pi. The range is [-0.5,0.5]. X-intercepts at multiples of pi.
    3.

    amplitude: 5; period: 2π; 2π; midline: y=0 y=0

    Two periods of a sine function, graphed over -2pi to 2pi. The range is [-5,5], amplitude of 5, period of 2pi.
    5.

    amplitude: 1; period: 2π; 2π; midline: y=1 y=1

    A graph of two periods of a cosine function, graphed over -7pi/3 to 5pi/3. Range is [0,2], Period is 2pi, amplitude is1.
    7.

    amplitude: 3; period: 6π; 6π; midline: y=0 y=0

    A graph of two periods of a cosine function, over -7pi/2 to 17pi/2. The range is [-3,3], period is 6pi, and amplitude is 3.
    9.

    amplitude: none; period: π; π; midline: y=0, y=0, asymptotes: x= 2π 3 +πk, x= 2π 3 +πk, where k k is an integer

    A graph of two periods of a tangent function over -5pi/6 to 7pi/6. Period is pi, midline at y=0.
    11.

    amplitude: none; period: 2π 3 ; 2π 3 ; midline: y=0, y=0, asymptotes: x= π 3 k, x= π 3 k, where k k is an integer

    A graph of two periods of a cosecant functinon, over -2pi/3 to 2pi/3. Vertical asymptotes at multiples of pi/3. Period of 2pi/3.
    13.

    amplitude: none; period: 2π; 2π; midline: y=3 y=3

    A graph of two periods of a cosecant function, graphed from -9pi/4 to 7pi/4. Period is 2pi, midline at y=-3.
    15.

    amplitude: 2; period: 2; midline: y=0; y=0; f( x )=2sin( π( x1 ) ) f( x )=2sin( π( x1 ) )

    17.

    amplitude: 1; period: 12; phase shift: −6; −6; midline y=−3 y=−3

    19.

    D( t )=6812sin( π 12 x ) D( t )=6812sin( π 12 x )

    21.

    period: π 6 ; π 6 ; horizontal shift: −7 −7

    23.

    f( x )=sec( πx ); f( x )=sec( πx ); period: 2; phase shift: 0

    25.

    4 4

    27.

    The views are different because the period of the wave is 1 25 . 1 25 . Over a bigger domain, there will be more cycles of the graph.

    Two side-by-side graphs of a sinusodial function. The first graph is graphed over 0 to 1, the second graph is graphed over 0 to 3. There are many periods for each.
    29.

    3 5 3 5

    31.

    On the approximate intervals ( 0.5,1 ),( 1.6,2.1 ),( 2.6,3.1 ),( 3.7,4.2 ),( 4.7,5.2 ),(5.6,6.28) ( 0.5,1 ),( 1.6,2.1 ),( 2.6,3.1 ),( 3.7,4.2 ),( 4.7,5.2 ),(5.6,6.28)

    33.

    f( x )=2cos( 12( x+ π 4 ) )+3 f( x )=2cos( 12( x+ π 4 ) )+3

    A graph of one period of a cosine function, graphed over -pi/4 to 0. Range is [1,5], period is pi/6.
    35.

    This graph is periodic with a period of 2π. 2π.

    A graph of two periods of a sinusoidal function, The graph has a period of 2pi.
    37.

    π 3 π 3

    39.

    π 2 π 2

    41.

    1 ( 12x ) 2 1 ( 12x ) 2

    43.

    1 1+ x 4 1 1+ x 4

    45.

    x+1 x x+1 x

    47.

    False

    49.

    approximately 0.07 radians


    13.2.6: Chapter 6 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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