# 5.3 The other trigonometric functions  (Page 7/13)

 Page 7 / 13

## Algebraic

For the following exercises, find the exact value of each expression.

$\mathrm{tan}\text{\hspace{0.17em}}\frac{\pi }{6}$

$\mathrm{sec}\text{\hspace{0.17em}}\frac{\pi }{6}$

$\frac{2\sqrt{3}}{3}$

$\mathrm{csc}\text{\hspace{0.17em}}\frac{\pi }{6}$

$\mathrm{cot}\text{\hspace{0.17em}}\frac{\pi }{6}$

$\sqrt{3}$

$\mathrm{tan}\text{\hspace{0.17em}}\frac{\pi }{4}$

$\mathrm{sec}\text{\hspace{0.17em}}\frac{\pi }{4}$

$\sqrt{2}$

$\mathrm{csc}\text{\hspace{0.17em}}\frac{\pi }{4}$

$\mathrm{cot}\text{\hspace{0.17em}}\frac{\pi }{4}$

1

$\mathrm{tan}\text{\hspace{0.17em}}\frac{\pi }{3}$

$\mathrm{sec}\text{\hspace{0.17em}}\frac{\pi }{3}$

2

$\mathrm{csc}\text{\hspace{0.17em}}\frac{\pi }{3}$

$\mathrm{cot}\text{\hspace{0.17em}}\frac{\pi }{3}$

$\frac{\sqrt{3}}{3}$

For the following exercises, use reference angles to evaluate the expression.

$\mathrm{tan}\text{\hspace{0.17em}}\frac{5\pi }{6}$

$\mathrm{sec}\text{\hspace{0.17em}}\frac{7\pi }{6}$

$-\frac{2\sqrt{3}}{3}$

$\mathrm{csc}\text{\hspace{0.17em}}\frac{11\pi }{6}$

$\mathrm{cot}\text{\hspace{0.17em}}\frac{13\pi }{6}$

$\sqrt{3}$

$\mathrm{tan}\text{\hspace{0.17em}}\frac{7\pi }{4}$

$\mathrm{sec}\text{\hspace{0.17em}}\frac{3\pi }{4}$

$-\sqrt{2}$

$\mathrm{csc}\text{\hspace{0.17em}}\frac{5\pi }{4}$

$\mathrm{cot}\text{\hspace{0.17em}}\frac{11\pi }{4}$

−1

$\mathrm{tan}\text{\hspace{0.17em}}\frac{8\pi }{3}$

$\mathrm{sec}\text{\hspace{0.17em}}\frac{4\pi }{3}$

−2

$\mathrm{csc}\text{\hspace{0.17em}}\frac{2\pi }{3}$

$\mathrm{cot}\text{\hspace{0.17em}}\frac{5\pi }{3}$

$-\frac{\sqrt{3}}{3}$

$\mathrm{tan}\text{\hspace{0.17em}}225°$

$\mathrm{sec}\text{\hspace{0.17em}}300°$

2

$\mathrm{csc}\text{\hspace{0.17em}}150°$

$\mathrm{cot}\text{\hspace{0.17em}}240°$

$\frac{\sqrt{3}}{3}$

$\mathrm{tan}\text{\hspace{0.17em}}330°$

$\mathrm{sec}\text{\hspace{0.17em}}120°$

−2

$\mathrm{csc}\text{\hspace{0.17em}}210°$

$\mathrm{cot}\text{\hspace{0.17em}}315°$

−1

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 $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}40°\approx 0.643\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}40°\approx 0.766\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{sec}\text{\hspace{0.17em}}40°,\text{csc}\text{\hspace{0.17em}}40°,\text{tan}\text{\hspace{0.17em}}40°,\text{and}\text{\hspace{0.17em}}\text{cot}\text{\hspace{0.17em}}40°.$

If $\text{\hspace{0.17em}}\text{sin}\text{\hspace{0.17em}}t=\frac{\sqrt{2}}{2},$ what is the $\text{\hspace{0.17em}}\text{sin}\left(-t\right)?$

$\text{\hspace{0.17em}}-\frac{\sqrt{2}}{2}\text{\hspace{0.17em}}$

If $\text{\hspace{0.17em}}\text{cos}\text{\hspace{0.17em}}t=\frac{1}{2},$ what is the $\text{\hspace{0.17em}}\text{cos}\left(-t\right)?$

If $\text{\hspace{0.17em}}\text{sec}\text{\hspace{0.17em}}t=3.1,$ what is the $\text{\hspace{0.17em}}\text{sec}\left(-t\right)?$

3.1

If $\text{\hspace{0.17em}}\text{csc}\text{\hspace{0.17em}}t=0.34,$ what is the $\text{\hspace{0.17em}}\text{csc}\left(-t\right)?$

If $\text{\hspace{0.17em}}\text{tan}\text{\hspace{0.17em}}t=-1.4,$ what is the $\text{\hspace{0.17em}}\text{tan}\left(-t\right)?$

1.4

If $\text{\hspace{0.17em}}\text{cot}\text{\hspace{0.17em}}t=9.23,$ what is the $\text{\hspace{0.17em}}\text{cot}\left(-t\right)?$

## Graphical

For the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions.

$\mathrm{sin}\text{\hspace{0.17em}}t=\frac{\sqrt{2}}{2},\mathrm{cos}\text{\hspace{0.17em}}t=\frac{\sqrt{2}}{2},\mathrm{tan}\text{\hspace{0.17em}}t=1,\mathrm{cot}\text{\hspace{0.17em}}t=1,\mathrm{sec}\text{\hspace{0.17em}}t=\sqrt{2},\mathrm{csc}\text{\hspace{0.17em}}t=\sqrt{2}$

$\mathrm{sin}\text{\hspace{0.17em}}t=-\frac{\sqrt{3}}{2},\mathrm{cos}\text{\hspace{0.17em}}t=-\frac{1}{2},\mathrm{tan}\text{\hspace{0.17em}}t=\sqrt{3},\mathrm{cot}\text{\hspace{0.17em}}t=\frac{\sqrt{3}}{3},\mathrm{sec}\text{\hspace{0.17em}}t=-2,\mathrm{csc}\text{\hspace{0.17em}}t=-\frac{2\sqrt{3}}{3}$

## Technology

For the following exercises, use a graphing calculator to evaluate.

$\mathrm{csc}\text{\hspace{0.17em}}\frac{5\pi }{9}$

$\mathrm{cot}\text{\hspace{0.17em}}\frac{4\pi }{7}$

–0.228

$\mathrm{sec}\text{\hspace{0.17em}}\frac{\pi }{10}$

$\mathrm{tan}\text{\hspace{0.17em}}\frac{5\pi }{8}$

–2.414

$\mathrm{sec}\text{\hspace{0.17em}}\frac{3\pi }{4}$

$\mathrm{csc}\text{\hspace{0.17em}}\frac{\pi }{4}$

1.414

$\text{tan}\text{\hspace{0.17em}}98°$

$\mathrm{cot}\text{\hspace{0.17em}}33°$

1.540

$\mathrm{cot}\text{\hspace{0.17em}}140°$

$\mathrm{sec}\text{\hspace{0.17em}}310°$

1.556

## Extensions

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 $\text{\hspace{0.17em}}f\left(x\right)=2\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is even, odd, or neither.

Determine whether the function $f\left(x\right)=3{\mathrm{sin}}^{2}x\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\mathrm{sec}\text{\hspace{0.17em}}x$ is even, odd, or neither.

even

Determine whether the function $f\left(x\right)=\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}-2{\mathrm{cos}}^{2}x$ is even, odd, or neither.

Determine whether the function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{csc}}^{2}x+\mathrm{sec}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is even, odd, or neither.

even

For the following exercises, use identities to simplify the expression.

$\mathrm{csc}\text{\hspace{0.17em}}t\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}t$

$\frac{\mathrm{sec}\text{\hspace{0.17em}}t}{\mathrm{csc}\text{\hspace{0.17em}}t}$

$\frac{\mathrm{sin}\text{\hspace{0.17em}}t}{\mathrm{cos}\text{\hspace{0.17em}}t}=\mathrm{tan}\text{\hspace{0.17em}}t$

## Real-world applications

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 $\text{\hspace{0.17em}}55°.\text{\hspace{0.17em}}$

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 $\text{\hspace{0.17em}}55°.\text{\hspace{0.17em}}$

how can are find the domain and range of a relations
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
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Robert
can I see the picture
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with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
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SLIMANE
What is domain
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Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
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