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How can the graph of $\text{\hspace{0.17em}}y=\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ be used to construct the graph of $\text{\hspace{0.17em}}y=\mathrm{sec}\text{\hspace{0.17em}}x?$
Explain why the period of $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is equal to $\text{\hspace{0.17em}}\pi .$
Answers will vary. Using the unit circle, one can show that $\text{\hspace{0.17em}}\mathrm{tan}\left(x+\pi \right)=\mathrm{tan}\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$
Why are there no intercepts on the graph of $\text{\hspace{0.17em}}y=\mathrm{csc}\text{\hspace{0.17em}}x?$
How does the period of $\text{\hspace{0.17em}}y=\mathrm{csc}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ compare with the period of $\text{\hspace{0.17em}}y=\mathrm{sin}\text{\hspace{0.17em}}x?$
The period is the same: $\text{\hspace{0.17em}}2\pi .$
For the following exercises, match each trigonometric function with one of the following graphs.
$f\left(x\right)=\mathrm{tan}\text{\hspace{0.17em}}x$
$f\left(x\right)=\mathrm{csc}\text{\hspace{0.17em}}x$
For the following exercises, find the period and horizontal shift of each of the functions.
$f\left(x\right)=2\mathrm{tan}\left(4x-32\right)$
$h\left(x\right)=2\mathrm{sec}\left(\frac{\pi}{4}\left(x+1\right)\right)$
period: 8; horizontal shift: 1 unit to left
$m\left(x\right)=6\mathrm{csc}\left(\frac{\pi}{3}x+\pi \right)$
If $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x=-1.5,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}\mathrm{tan}\left(-x\right).$
1.5
If $\text{\hspace{0.17em}}\mathrm{sec}\text{\hspace{0.17em}}x=2,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}\mathrm{sec}\left(-x\right).$
If $\text{\hspace{0.17em}}\mathrm{csc}\text{\hspace{0.17em}}x=-5,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}\mathrm{csc}\left(-x\right).$
5
If $\text{\hspace{0.17em}}x\mathrm{sin}\text{\hspace{0.17em}}x=2,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}\left(-x\right)\mathrm{sin}\left(-x\right).$
For the following exercises, rewrite each expression such that the argument $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is positive.
$\mathrm{cot}\left(-x\right)\mathrm{cos}\left(-x\right)+\mathrm{sin}\left(-x\right)$
$-\mathrm{cot}x\mathrm{cos}x-\mathrm{sin}x$
$\mathrm{cos}\left(-x\right)+\mathrm{tan}\left(-x\right)\mathrm{sin}\left(-x\right)$
For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.
$f\left(x\right)=2\mathrm{tan}\left(4x-32\right)$
stretching factor: 2; period: $\text{}\frac{\pi}{4};\text{}$ asymptotes: $\text{}x=\frac{1}{4}\left(\frac{\pi}{2}+\pi k\right)+8,\text{where}k\text{isaninteger}$
$\text{\hspace{0.17em}}h\left(x\right)=2\mathrm{sec}\left(\frac{\pi}{4}\left(x+1\right)\right)\text{\hspace{0.17em}}$
$m\left(x\right)=6\mathrm{csc}\left(\frac{\pi}{3}x+\pi \right)$
stretching factor: 6; period: 6; asymptotes: $\text{}x=3k,\text{where}k\text{isaninteger}$
$j\left(x\right)=\mathrm{tan}\left(\frac{\pi}{2}x\right)$
$p(x)=\mathrm{tan}\left(x-\frac{\pi}{2}\right)$
stretching factor: 1; period: $\text{}\pi ;\text{}$ asymptotes: $\text{}x=\pi k,\text{where}k\text{isaninteger}$
$f(x)=4\mathrm{tan}(x)$
$f(x)=\mathrm{tan}\left(x+\frac{\pi}{4}\right)$
Stretching factor: 1; period: $\text{}\pi ;\text{}$ asymptotes: $\text{}x=\frac{\pi}{4}+\pi k,\text{where}k\text{isaninteger}$
$f(x)=\pi \mathrm{tan}\left(\pi x-\pi \right)-\pi $
$f\left(x\right)=2\mathrm{csc}\left(x\right)$
stretching factor: 2; period: $\text{}2\pi ;\text{}$ asymptotes: $\text{}x=\pi k,\text{where}k\text{isaninteger}$
$f\left(x\right)=-\frac{1}{4}\mathrm{csc}\left(x\right)$
$f(x)=4\mathrm{sec}\left(3x\right)$
stretching factor: 4; period: $\text{}\frac{2\pi}{3};\text{}$ asymptotes: $\text{}x=\frac{\pi}{6}k,\text{where}k\text{isanoddinteger}$
$f(x)=-3\mathrm{cot}\left(2x\right)$
$f(x)=7\mathrm{sec}\left(5x\right)$
stretching factor: 7; period: $\text{}\frac{2\pi}{5};\text{}$ asymptotes: $\text{}x=\frac{\pi}{10}k,\text{where}k\text{isanoddinteger}$
$f(x)=\frac{9}{10}\mathrm{csc}\left(\pi x\right)$
$f(x)=2\mathrm{csc}\left(x+\frac{\pi}{4}\right)-1$
stretching factor: 2; period: $\text{}2\pi ;\text{}$ asymptotes: $\text{}x=-\frac{\pi}{4}+\pi k,\text{where}k\text{isaninteger}$
$f(x)=-\mathrm{sec}\left(x-\frac{\pi}{3}\right)-2$
$f(x)=\frac{7}{5}\mathrm{csc}\left(x-\frac{\pi}{4}\right)$
stretching factor: $\text{}\frac{7}{5};\text{}$ period: $\text{}2\pi ;\text{}$ asymptotes: $\text{}x=\frac{\pi}{4}+\pi k,\text{where}k\text{isaninteger}$
$f(x)=5\left(\mathrm{cot}\left(x+\frac{\pi}{2}\right)-3\right)$
For the following exercises, find and graph two periods of the periodic function with the given stretching factor, $\text{\hspace{0.17em}}\left|A\right|,\text{\hspace{0.17em}}$ period, and phase shift.
A tangent curve, $\text{\hspace{0.17em}}A=1,\text{\hspace{0.17em}}$ period of $\text{\hspace{0.17em}}\frac{\pi}{3};\text{\hspace{0.17em}}$ and phase shift $\text{\hspace{0.17em}}\left(h,\text{\hspace{0.17em}}k\right)=\left(\frac{\pi}{4},2\right)$
$y=\mathrm{tan}\left(3\left(x-\frac{\pi}{4}\right)\right)+2$
A tangent curve, $\text{\hspace{0.17em}}A=\mathrm{-2},\text{\hspace{0.17em}}$ period of $\text{\hspace{0.17em}}\frac{\pi}{4},\text{\hspace{0.17em}}$ and phase shift $\text{\hspace{0.17em}}\left(h,\text{\hspace{0.17em}}k\right)=\left(-\frac{\pi}{4},\text{\hspace{0.17em}}\mathrm{-2}\right)$
For the following exercises, find an equation for the graph of each function.
For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input $\text{\hspace{0.17em}}\mathrm{csc}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ as $\text{\hspace{0.17em}}\frac{1}{\mathrm{sin}\text{\hspace{0.17em}}x}.$
$f(x)=\left|\mathrm{csc}\left(x\right)\right|$
$f(x)={2}^{\mathrm{csc}\left(x\right)}$
$f(x)=\frac{\mathrm{csc}\left(x\right)}{\mathrm{sec}\left(x\right)}$
Graph $\text{\hspace{0.17em}}f(x)=1+{\mathrm{sec}}^{2}\left(x\right)-{\mathrm{tan}}^{2}\left(x\right).\text{\hspace{0.17em}}$ What is the function shown in the graph?
$f(x)=\mathrm{cot}\left(100\pi x\right)$
The function $\text{\hspace{0.17em}}f\left(x\right)=20\mathrm{tan}\left(\frac{\pi}{10}x\right)\text{\hspace{0.17em}}$ marks the distance in the movement of a light beam from a police car across a wall for time $\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ in seconds, and distance $\text{\hspace{0.17em}}f\left(x\right),$ in feet.
Standing on the shore of a lake, a fisherman sights a boat far in the distance to his left. Let $\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ measured in radians, be the angle formed by the line of sight to the ship and a line due north from his position. Assume due north is 0 and $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is measured negative to the left and positive to the right. (See [link] .) The boat travels from due west to due east and, ignoring the curvature of the Earth, the distance $\text{\hspace{0.17em}}d\left(x\right),\text{\hspace{0.17em}}$ in kilometers, from the fisherman to the boat is given by the function $\text{\hspace{0.17em}}d\left(x\right)=1.5\mathrm{sec}\left(x\right).$
A laser rangefinder is locked on a comet approaching Earth. The distance $\text{\hspace{0.17em}}g\left(x\right),\text{\hspace{0.17em}}$ in kilometers, of the comet after $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ days, for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ in the interval 0 to 30 days, is given by $\text{\hspace{0.17em}}g\left(x\right)=\mathrm{250,000}\mathrm{csc}\left(\frac{\pi}{30}x\right).$
A video camera is focused on a rocket on a launching pad 2 miles from the camera. The angle of elevation from the ground to the rocket after $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ seconds is $\text{\hspace{0.17em}}\frac{\pi}{120}x.$
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