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

The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
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
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
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).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations