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For the following exercises, find the exact value of each expression.
$\mathrm{tan}\text{\hspace{0.17em}}\frac{\pi}{6}$
$\mathrm{sec}\text{\hspace{0.17em}}\frac{\pi}{6}$
$\frac{2\sqrt{3}}{3}$
$\mathrm{csc}\text{\hspace{0.17em}}\frac{\pi}{6}$
$\mathrm{cot}\text{\hspace{0.17em}}\frac{\pi}{6}$
$\sqrt{3}$
$\mathrm{tan}\text{\hspace{0.17em}}\frac{\pi}{4}$
$\mathrm{sec}\text{\hspace{0.17em}}\frac{\pi}{4}$
$\sqrt{2}$
$\mathrm{csc}\text{\hspace{0.17em}}\frac{\pi}{4}$
$\mathrm{tan}\text{\hspace{0.17em}}\frac{\pi}{3}$
$\mathrm{csc}\text{\hspace{0.17em}}\frac{\pi}{3}$
$\mathrm{cot}\text{\hspace{0.17em}}\frac{\pi}{3}$
$\frac{\sqrt{3}}{3}$
For the following exercises, use reference angles to evaluate the expression.
$\mathrm{tan}\text{\hspace{0.17em}}\frac{5\pi}{6}$
$\mathrm{sec}\text{\hspace{0.17em}}\frac{7\pi}{6}$
$-\frac{2\sqrt{3}}{3}$
$\mathrm{csc}\text{\hspace{0.17em}}\frac{11\pi}{6}$
$\mathrm{cot}\text{\hspace{0.17em}}\frac{13\pi}{6}$
$\sqrt{3}$
$\mathrm{tan}\text{\hspace{0.17em}}\frac{7\pi}{4}$
$\mathrm{sec}\text{\hspace{0.17em}}\frac{3\pi}{4}$
$-\sqrt{2}$
$\mathrm{csc}\text{\hspace{0.17em}}\frac{5\pi}{4}$
$\mathrm{tan}\text{\hspace{0.17em}}\frac{8\pi}{3}$
$\mathrm{csc}\text{\hspace{0.17em}}\frac{2\pi}{3}$
$\mathrm{cot}\text{\hspace{0.17em}}\frac{5\pi}{3}$
$-\frac{\sqrt{3}}{3}$
$\mathrm{tan}\text{\hspace{0.17em}}\mathrm{225\xb0}$
$\mathrm{csc}\text{\hspace{0.17em}}\mathrm{150\xb0}$
$\mathrm{cot}\text{\hspace{0.17em}}\mathrm{240\xb0}$
$\frac{\sqrt{3}}{3}$
$\mathrm{tan}\text{\hspace{0.17em}}\mathrm{330\xb0}$
$\mathrm{csc}\text{\hspace{0.17em}}\mathrm{210\xb0}$
If $\text{\hspace{0.17em}}\text{sin}\text{\hspace{0.17em}}t=\frac{3}{4},$ and $\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is in quadrant II, find $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}t,\mathrm{sec}\text{\hspace{0.17em}}t,\mathrm{csc}\text{\hspace{0.17em}}t,\mathrm{tan}\text{\hspace{0.17em}}t,\mathrm{cot}\text{\hspace{0.17em}}t.$
If $\text{\hspace{0.17em}}\text{cos}\text{\hspace{0.17em}}t=-\frac{1}{3},$ and $\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is in quadrant III, find $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}t,\mathrm{sec}\text{\hspace{0.17em}}t,\mathrm{csc}\text{\hspace{0.17em}}t,\mathrm{tan}\text{\hspace{0.17em}}t,\mathrm{cot}\text{\hspace{0.17em}}t.$
If $\mathrm{sin}\text{\hspace{0.17em}}t=-\frac{2\sqrt{2}}{3},\mathrm{sec}\text{\hspace{0.17em}}t=-3,\mathrm{csc}\text{\hspace{0.17em}}t=-\frac{3\sqrt{2}}{4},\mathrm{tan}\text{\hspace{0.17em}}t=2\sqrt{2},\mathrm{cot}\text{\hspace{0.17em}}t=\frac{\sqrt{2}}{4}$
If $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}t=\frac{12}{5},$ and $\text{\hspace{0.17em}}0\le t<\frac{\pi}{2},$ find $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}t,\mathrm{cos}\text{\hspace{0.17em}}t,\mathrm{sec}\text{\hspace{0.17em}}t,\mathrm{csc}\text{\hspace{0.17em}}t,$ and $\text{\hspace{0.17em}}\mathrm{cot}\text{\hspace{0.17em}}t.$
If $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}t=\frac{\sqrt{3}}{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}t=\frac{1}{2},$ find $\text{\hspace{0.17em}}\mathrm{sec}\text{\hspace{0.17em}}t,\mathrm{csc}\text{\hspace{0.17em}}t,\mathrm{tan}\text{\hspace{0.17em}}t,$ and $\text{\hspace{0.17em}}\mathrm{cot}\text{\hspace{0.17em}}t.$
$\mathrm{sec}\text{\hspace{0.17em}}t=2,\mathrm{csc}\text{\hspace{0.17em}}t=\frac{2\sqrt{3}}{3},\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}t=\sqrt{3},\text{\hspace{0.17em}}\mathrm{cot}\text{\hspace{0.17em}}t=\frac{\sqrt{3}}{3}$
If $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\mathrm{40\xb0}\approx 0.643\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\mathrm{40\xb0}\approx 0.766\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{sec}\text{\hspace{0.17em}}\mathrm{40\xb0},\text{csc}\text{\hspace{0.17em}}\mathrm{40\xb0},\text{tan}\text{\hspace{0.17em}}\mathrm{40\xb0},\text{and}\text{\hspace{0.17em}}\text{cot}\text{\hspace{0.17em}}\mathrm{40\xb0}.$
If $\text{\hspace{0.17em}}\text{sin}\text{\hspace{0.17em}}t=\frac{\sqrt{2}}{2},$ what is the $\text{\hspace{0.17em}}\text{sin}(-t)?$
$\text{\hspace{0.17em}}-\frac{\sqrt{2}}{2}\text{\hspace{0.17em}}$
If $\text{\hspace{0.17em}}\text{cos}\text{\hspace{0.17em}}t=\frac{1}{2},$ what is the $\text{\hspace{0.17em}}\text{cos}(-t)?$
If $\text{\hspace{0.17em}}\text{sec}\text{\hspace{0.17em}}t=3.1,$ what is the $\text{\hspace{0.17em}}\text{sec}(-t)?$
3.1
If $\text{\hspace{0.17em}}\text{csc}\text{\hspace{0.17em}}t=0.34,$ what is the $\text{\hspace{0.17em}}\text{csc}(-t)?$
If $\text{\hspace{0.17em}}\text{tan}\text{\hspace{0.17em}}t=-1.4,$ what is the $\text{\hspace{0.17em}}\text{tan}(-t)?$
1.4
If $\text{\hspace{0.17em}}\text{cot}\text{\hspace{0.17em}}t=9.23,$ what is the $\text{\hspace{0.17em}}\text{cot}(-t)?$
For the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions.
$\mathrm{sin}\text{\hspace{0.17em}}t=\frac{\sqrt{2}}{2},\mathrm{cos}\text{\hspace{0.17em}}t=\frac{\sqrt{2}}{2},\mathrm{tan}\text{\hspace{0.17em}}t=1,\mathrm{cot}\text{\hspace{0.17em}}t=1,\mathrm{sec}\text{\hspace{0.17em}}t=\sqrt{2},\mathrm{csc}\text{\hspace{0.17em}}t=\sqrt{2}$
$\mathrm{sin}\text{\hspace{0.17em}}t=-\frac{\sqrt{3}}{2},\mathrm{cos}\text{\hspace{0.17em}}t=-\frac{1}{2},\mathrm{tan}\text{\hspace{0.17em}}t=\sqrt{3},\mathrm{cot}\text{\hspace{0.17em}}t=\frac{\sqrt{3}}{3},\mathrm{sec}\text{\hspace{0.17em}}t=-2,\mathrm{csc}\text{\hspace{0.17em}}t=-\frac{2\sqrt{3}}{3}$
For the following exercises, use a graphing calculator to evaluate.
$\mathrm{csc}\text{\hspace{0.17em}}\frac{5\pi}{9}$
$\mathrm{sec}\text{\hspace{0.17em}}\frac{\pi}{10}$
$\mathrm{sec}\text{\hspace{0.17em}}\frac{3\pi}{4}$
$\text{tan}\text{\hspace{0.17em}}\mathrm{98\xb0}$
$\mathrm{cot}\text{\hspace{0.17em}}\mathrm{140\xb0}$
For the following exercises, use identities to evaluate the expression.
If $\text{\hspace{0.17em}}\mathrm{tan}\left(t\right)\approx 2.7,$ and $\text{\hspace{0.17em}}\mathrm{sin}\left(t\right)\approx 0.94,$ find $\text{\hspace{0.17em}}\mathrm{cos}\left(t\right).$
If $\text{\hspace{0.17em}}\mathrm{tan}\left(t\right)\approx 1.3,$ and $\text{\hspace{0.17em}}\mathrm{cos}\left(t\right)\approx 0.61,$ find $\text{\hspace{0.17em}}\mathrm{sin}\left(t\right).\text{\hspace{0.17em}}$
$\mathrm{sin}\left(t\right)\approx 0.79$
If $\text{\hspace{0.17em}}\mathrm{csc}\left(t\right)\approx 3.2,$ and $\text{\hspace{0.17em}}\mathrm{cos}\left(t\right)\approx 0.95,$ find $\text{\hspace{0.17em}}\mathrm{tan}\left(t\right).$
If $\text{\hspace{0.17em}}\mathrm{cot}\left(t\right)\approx 0.58,$ and $\text{\hspace{0.17em}}\mathrm{cos}\left(t\right)\approx 0.5,$ find $\text{\hspace{0.17em}}\mathrm{csc}\left(t\right).$
$\mathrm{csc}t\approx 1.16$
Determine whether the function $\text{\hspace{0.17em}}f(x)=2\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is even, odd, or neither.
Determine whether the function $f(x)=3{\mathrm{sin}}^{2}x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\mathrm{sec}\text{\hspace{0.17em}}x$ is even, odd, or neither.
even
Determine whether the function $f(x)=\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}-2{\mathrm{cos}}^{2}x$ is even, odd, or neither.
Determine whether the function $\text{\hspace{0.17em}}f(x)={\mathrm{csc}}^{2}x+\mathrm{sec}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is even, odd, or neither.
even
For the following exercises, use identities to simplify the expression.
$\mathrm{csc}\text{\hspace{0.17em}}t\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}t$
$\frac{\mathrm{sec}\text{\hspace{0.17em}}t}{\mathrm{csc}\text{\hspace{0.17em}}t}$
$\frac{\mathrm{sin}\text{\hspace{0.17em}}t}{\mathrm{cos}\text{\hspace{0.17em}}t}=\mathrm{tan}\text{\hspace{0.17em}}t$
The amount of sunlight in a certain city can be modeled by the function $\text{\hspace{0.17em}}h=15\mathrm{cos}\left(\frac{1}{600}d\right),$ where $\text{\hspace{0.17em}}h\text{\hspace{0.17em}}$ represents the hours of sunlight, and $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ is the day of the year. Use the equation to find how many hours of sunlight there are on February 10, the 42 ^{nd} day of the year. State the period of the function.
The amount of sunlight in a certain city can be modeled by the function $\text{\hspace{0.17em}}h=16\mathrm{cos}\left(\frac{1}{500}d\right),$ where $\text{\hspace{0.17em}}h\text{\hspace{0.17em}}$ represents the hours of sunlight, and $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ is the day of the year. Use the equation to find how many hours of sunlight there are on September 24, the 267 ^{th} day of the year. State the period of the function.
13.77 hours, period: $\text{\hspace{0.17em}}1000\pi \text{\hspace{0.17em}}$
The equation $\text{\hspace{0.17em}}P=20\mathrm{sin}\left(2\pi t\right)+100\text{\hspace{0.17em}}$ models the blood pressure, $\text{\hspace{0.17em}}P,$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ represents time in seconds. (a) Find the blood pressure after 15 seconds. (b) What are the maximum and minimum blood pressures?
The height of a piston, $\text{\hspace{0.17em}}h,$ in inches, can be modeled by the equation $\text{\hspace{0.17em}}y=2\mathrm{cos}\text{\hspace{0.17em}}x+6,$ where $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ represents the crank angle. Find the height of the piston when the crank angle is $\text{\hspace{0.17em}}\mathrm{55\xb0}.\text{\hspace{0.17em}}$
7.73 inches
The height of a piston, $\text{\hspace{0.17em}}h,$ in inches, can be modeled by the equation $\text{\hspace{0.17em}}y=2\mathrm{cos}\text{\hspace{0.17em}}x+5,$ where $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ represents the crank angle. Find the height of the piston when the crank angle is $\text{\hspace{0.17em}}\mathrm{55\xb0}.\text{\hspace{0.17em}}$
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