13.2.6: Chapter 6

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Try It

6.1 Graphs of the Sine and Cosine Functions

$6 π$

$\frac{1}{2}$ compressed

$\frac{π}{2};$ right

2 units up

midline: $y = 0;$ amplitude: $| A | = \frac{1}{2};$ period: $P = \frac{2 π}{| B |} = 6 π;$ phase shift: $\frac{C}{B} = π$

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

two possibilities: $y = 4 \sin (\frac{π}{5} x - \frac{π}{5}) + 4$ or $y = - 4 \sin (\frac{π}{5} x + \frac{4 π}{5}) + 4$

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

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

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

midline: $y = 0;$ amplitude: $| A | = 2;$ period: $P = \frac{2 π}{| B |} = 6;$ phase shift: $\frac{C}{B} = - \frac{1}{2}$

10.

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

11.

$y = 3 \cos (x) - 4$

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

6.2 Graphs of the Other Trigonometric Functions

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

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

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

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

$A graph of one period of a modified secant function, which looks like an downward facing prarbola and a upward facing parabola.$

$A graph of one period of a modified secant function. There are two vertical asymptotes, one at approximately x=-pi/20 and one approximately at 3pi/16.$

$A graph of one period of a modified secant function, which looks like an downward facing prarbola and a upward facing parabola.$

$A graph of two periods of both a secant and consine function. Grpah shows that cosine function has local maximums where secant function has local minimums and vice versa.$

6.3 Inverse Trigonometric Functions

$\arccos (0.8776) \approx 0.5$

ⓐ $- \frac{π}{2};$
ⓑ $- \frac{π}{4};$
ⓒ $π;$
ⓓ $\frac{π}{3}$

1.9823 or 113.578°

$\sin^{−1} (0.6) = 36.87° = 0.6435$ radians

$\frac{π}{8}; \frac{2 π}{9}$

$\frac{3 π}{4}$

$\frac{12}{13}$

$\frac{4 \sqrt{2}}{9}$

$\frac{4 x}{\sqrt{16 x^{2} + 1}}$

6.1 Section Exercises

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

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

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

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

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

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

amplitude: 4; period: $2 π;$ midline: $y = 0;$ maximum $y = 4$ occurs at $x = \frac{π}{2};$ minimum: $y = - 4$ occurs at $x = \frac{3 π}{2};$ one full period occurs from $x = 0$ to $x = 2 π$

11.

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

amplitude: 1; period: $π;$ midline: $y = 0;$ maximum: $y = 1$ occurs at $x = π;$ minimum: $y = - 1$ occurs at $x = \frac{π}{2};$ one full period is graphed from $x = 0$ to $x = π$

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: $y = 0;$ maximum: $y = 4$ occurs at $x = 0;$ minimum: $y = - 4$ occurs at $x = 1$

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: $\frac{π}{4};$ midline: $y = 5;$ maximum: $y = 8$ occurs at $x = 0.12;$ minimum: $y = 2$ occurs at $x = 0.516;$ horizontal shift: $- 4;$ vertical translation 5; one period occurs from $x = 0$ to $x = \frac{π}{4}$

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

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

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: $4 π;$ midline: $y = 0;$ maximum: $y = 1$ occurs at $t = 11.52;$ minimum: $y = - 1$ occurs at $t = 5.24;$ phase shift: $- \frac{10 π}{3};$ vertical shift: 0

23.

amplitude: 2; midline: $y = - 3;$ period: 4; equation: $f (x) = 2 \sin (\frac{π}{2} x) - 3$

25.

amplitude: 2; period: 5; midline: $y = 3;$ equation: $f (x) = - 2 \cos (\frac{2 π}{5} x) + 3$

27.

amplitude: 4; period: 2; midline: $y = 0;$ equation: $f (x) = - 4 \cos (π (x - \frac{π}{2}))$

29.

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

31.

$0, π$

33.

$sin (\frac{π}{2}) = 1$

35.

$\frac{π}{2}$

37.

$f (x) = sin x$ is symmetric

39.

$\frac{π}{3}, \frac{5 π}{3}$

41.

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

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 $h (x) = x + sin x$ will increase without bound as well. The graph is bounded between the graphs of $y = x + 1$ and $y = x - 1$ 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 $| x |$ grows. There appears to be a horizontal asymptote at $y = 0$ .

$24a9c053feb5ae1c3c33e956a80848a5c26e6d07$

6.2 Section Exercises

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

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

The period is the same: $2 π .$

III

11.

period: 8; horizontal shift: 1 unit to left

13.

1.5

15.

17.

$- \cot x \cos x - \sin x$

19.

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

stretching factor: 2; period: $\frac{π}{4};$ asymptotes: $x = \frac{1}{4} (\frac{π}{2} + π k) + 8, where k is an integer$

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: $x = 3 k, where k is an integer$

23.

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

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

25.

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

Stretching factor: 1; period: $π;$ asymptotes: $x = \frac{π}{4} + π k, where k is an integer$

27.

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

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

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: $\frac{2 π}{3};$ asymptotes: $x = \frac{π}{6} k, where k is an odd integer$

31.

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

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

33.

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

stretching factor: 2; period: $2 π;$ asymptotes: $x = - \frac{π}{4} + π k, where k is an integer$

35.

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

stretching factor: $\frac{7}{5};$ period: $2 π;$ asymptotes: $x = \frac{π}{4} + π k, where k is an integer$

37.

$y = \tan (3 (x - \frac{π}{4})) + 2$

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

39.

$f (x) = \csc (2 x)$

41.

$f (x) = \csc (4 x)$

43.

$f (x) = 2 \csc x$

45.

$f (x) = \frac{1}{2} \tan (100 π x)$

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.

ⓐ $(- \frac{π}{2}, \frac{π}{2});$
ⓑ $A graph of a half period of a secant function. Vertical asymptotes at x=-pi/2 and pi/2.$
ⓒ $x = - \frac{π}{2}$ and $x = \frac{π}{2};$ the distance grows without bound as $| x |$ approaches $\frac{π}{2}$ —i.e., at right angles to the line representing due north, the boat would be so far away, the fisherman could not see it;
ⓓ3; when $x = - \frac{π}{3},$ the boat is 3 km away;
ⓔ 1.73; when $x = \frac{π}{6},$ the boat is about 1.73 km away;
ⓕ 1.5 km; when $x = 0$

57.

ⓐ $h (x) = 2 \tan (\frac{π}{120} x);$
ⓑ $An exponentially increasing function with a vertical asymptote at x=60.$
ⓒ $h (0) = 0 :$ after 0 seconds, the rocket is 0 mi above the ground; $h (30) = 2 :$ after 30 seconds, the rockets is 2 mi high;
ⓓAs $x$ approaches 60 seconds, the values of $h (x)$ grow increasingly large. The distance to the rocket is growing so large that the camera can no longer track it.

6.3 Section Exercises

The function $y = \sin x$ is one-to-one on $[- \frac{π}{2}, \frac{π}{2}];$ thus, this interval is the range of the inverse function of $y = \sin x,$ $f (x) = \sin^{- 1} x .$ The function $y = \cos x$ is one-to-one on $[0, π];$ thus, this interval is the range of the inverse function of $y = \cos x, f (x) = \cos^{- 1} x .$

$\frac{π}{6}$ is the radian measure of an angle between $- \frac{π}{2}$ and $\frac{π}{2}$ whose sine is 0.5.

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 $[- \frac{π}{2}, \frac{π}{2}]$ so that it is one-to-one and possesses an inverse.

True . The angle, $θ_{1}$ that equals $\arccos (- x)$ , $x > 0$ , will be a second quadrant angle with reference angle, $θ_{2}$ , where $θ_{2}$ equals $\arccos x$ , $x > 0$ . Since $θ_{2}$ is the reference angle for $θ_{1}$ , $θ_{2} = π - θ_{1}$ and $\arccos (- x)$ = $π - \arccos x$ -

$- \frac{π}{6}$

11.

$\frac{3 π}{4}$

13.

$- \frac{π}{3}$

15.

$\frac{π}{3}$

17.

1.98

19.

0.93

21.

1.41

23.

0.56 radians

25.

27.

0.71

29.

-0.71

31.

$- \frac{π}{4}$

33.

0.8

35.

$\frac{5}{13}$

37.

$\frac{x - 1}{\sqrt{- x^{2} + 2 x}}$

39.

$\frac{\sqrt{x^{2} - 1}}{x}$

41.

$\frac{x + 0.5}{\sqrt{- x^{2} - x + \frac{3}{4}}}$

43.

$\frac{\sqrt{2 x + 1}}{x + 1}$

45.

$\frac{\sqrt{2 x + 1}}{x}$

47.

$t$

49.

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

domain $[- 1, 1];$ range $[0, π]$

51.

approximately $x = 0.00$

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

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

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

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

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

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

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

amplitude: 6; period: $\frac{2 π}{3};$ midline: $y = - 1;$ no asymptotes

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

stretching factor: none; period: $π;$ midline: $y = - 4;$ asymptotes: $x = \frac{π}{2} + π k,$ where $k$ 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: $\frac{π}{4};$ midline: $y = - 2;$ asymptotes: $x = \frac{π}{8} + \frac{π}{4} k,$ where $k$ 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: $2 π;$ no phase shift; asymptotes: $x = \frac{π}{2} k,$ where $k$ 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: $\frac{2 π}{5};$ no phase shift; asymptotes: $x = \frac{π}{5} k,$ where $k$ 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: $4 π;$ no phase shift; asymptotes: $x = 2 π k,$ where $k$ 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.

$\frac{π}{6}$

31.

$\frac{π}{4}$

33.

$\frac{π}{3}$

35.

No solution

37.

$\frac{12}{5}$

39.

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

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

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

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

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

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

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

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

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

amplitude: none; period: $π;$ midline: $y = 0,$ asymptotes: $x = \frac{2 π}{3} + π k,$ where $k$ 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: $\frac{2 π}{3};$ midline: $y = 0,$ asymptotes: $x = \frac{π}{3} k,$ where $k$ 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: $2 π;$ midline: $y = - 3$

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

15.

amplitude: 2; period: 2; midline: $y = 0;$ $f (x) = 2 \sin (π (x - 1))$

17.

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

19.

$D (t) = 68 - 12 \sin (\frac{π}{12} x)$

21.

period: $\frac{π}{6};$ horizontal shift: $−7$

23.

$f (x) = \sec (π x);$ period: 2; phase shift: 0

25.

$4$

27.

The views are different because the period of the wave is $\frac{1}{25} .$ 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.

$\frac{3}{5}$

31.

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

33.

$f (x) = 2 \cos (12 (x + \frac{π}{4})) + 3$

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

35.

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

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

37.

$\frac{π}{3}$

39.

$\frac{π}{2}$

41.

$\sqrt{1 - {(1 - 2 x)}^{2}}$

43.

$\frac{1}{\sqrt{1 + x^{4}}}$

45.

$\frac{x + 1}{x}$

47.

False

49.

approximately 0.07 radians

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6.1 Graphs of the Sine and Cosine Functions

6.2 Graphs of the Other Trigonometric Functions

6.3 Inverse Trigonometric Functions