# 7.4 The other trigonometric functions  (Page 8/14)

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Find the area of the sector of a circle with diameter 32 feet and an angle of $\text{\hspace{0.17em}}\frac{3\pi }{5}\text{\hspace{0.17em}}$ radians.

For the following exercises, find the angle between $\text{\hspace{0.17em}}0°\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\text{360°}\text{\hspace{0.17em}}$ that is coterminal with the given angle.

$420°$

$60°$

$-80°$

For the following exercises, find the angle between 0 and $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ in radians that is coterminal with the given angle.

$-\text{\hspace{0.17em}}\frac{20\pi }{11}$

$\frac{2\pi }{11}$

$\frac{14\pi }{5}$

For the following exercises, draw the angle provided in standard position on the Cartesian plane.

$-210°$

$75°$

$\frac{5\pi }{4}$

$-\frac{\pi }{3}$

Find the linear speed of a point on the equator of the earth if the earth has a radius of 3,960 miles and the earth rotates on its axis every 24 hours. Express answer in miles per hour. Round to the nearest hundredth.

1036.73 miles per hour

A car wheel with a diameter of 18 inches spins at the rate of 10 revolutions per second. What is the car's speed in miles per hour? Round to the nearest hundredth.

## Right Triangle Trigonometry

For the following exercises, use side lengths to evaluate.

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

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

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

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

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

$\mathrm{cos}\left(\frac{\pi }{2}\right)=\mathrm{sin}\left(___°\right)$

$\mathrm{csc}\left(18°\right)=\mathrm{sec}\left(___°\right)$

$72°$

For the following exercises, use the given information to find the lengths of the other two sides of the right triangle.

$\mathrm{cos}\text{\hspace{0.17em}}B=\frac{3}{5},a=6$

$\mathrm{tan}\text{\hspace{0.17em}}A=\frac{5}{9},b=6$

$a=\frac{10}{3},c=\frac{2\sqrt{106}}{3}$

For the following exercises, use [link] to evaluate each trigonometric function.

$\mathrm{tan}\text{\hspace{0.17em}}B$

$\frac{6}{11}$

For the following exercises, solve for the unknown sides of the given triangle.

$a=\frac{5\sqrt{3}}{2},b=\frac{5}{2}$

A 15-ft ladder leans against a building so that the angle between the ground and the ladder is $\text{\hspace{0.17em}}70°.\text{\hspace{0.17em}}$ How high does the ladder reach up the side of the building? Find the answer to four decimal places.

The angle of elevation to the top of a building in Baltimore is found to be 4 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building. Find the answer to four decimal places.

369.2136 ft

## Unit Circle

Find the exact value of $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\frac{\pi }{3}.$

Find the exact value of $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\frac{\pi }{4}.$

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

Find the exact value of $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\pi .$

State the reference angle for $\text{\hspace{0.17em}}300°.\text{\hspace{0.17em}}$

$60°$

State the reference angle for $\text{\hspace{0.17em}}\frac{3\pi }{4}.$

Compute cosine of $\text{\hspace{0.17em}}330°.$

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

Compute sine of $\text{\hspace{0.17em}}\frac{5\pi }{4}.$

State the domain of the sine and cosine functions.

all real numbers

State the range of the sine and cosine functions.

## The Other Trigonometric Functions

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

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

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

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

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

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

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

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

$\mathrm{sec}\text{\hspace{0.17em}}\frac{11\pi }{3}$

2

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

If $\text{\hspace{0.17em}}\mathrm{sec}\left(t\right)=-2.5,$ what is the $\text{\hspace{0.17em}}\text{sec}\left(-t\right)?$

–2.5

If $\text{\hspace{0.17em}}\text{tan}\left(t\right)=-0.6,$ what is the $\text{\hspace{0.17em}}\text{tan}\left(-t\right)?$

If $\text{\hspace{0.17em}}\text{tan}\left(t\right)=\frac{1}{3},$ find $\text{\hspace{0.17em}}\text{tan}\left(t-\pi \right).$

$\frac{1}{3}$

If $\text{\hspace{0.17em}}\text{cos}\left(t\right)=\frac{\sqrt{2}}{2},$ find $\text{\hspace{0.17em}}\text{sin}\left(t+2\pi \right).$

Which trigonometric functions are even?

cosine, secant

Which trigonometric functions are odd?

## Chapter practice test

Convert $\text{\hspace{0.17em}}\frac{5\pi }{6}\text{\hspace{0.17em}}$ radians to degrees.

$150°$

Convert $\text{\hspace{0.17em}}-620°\text{\hspace{0.17em}}$ to radians.

Find the length of a circular arc with a radius 12 centimeters subtended by the central angle of $\text{\hspace{0.17em}}30°.$

6.283 centimeters

Find the area of the sector with radius of 8 feet and an angle of $\text{\hspace{0.17em}}\frac{5\pi }{4}\text{\hspace{0.17em}}$ radians.

Find the angle between $\text{\hspace{0.17em}}0°\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\text{360°}\text{\hspace{0.17em}}$ that is coterminal with $\text{\hspace{0.17em}}375°.$

$15°$

Find the angle between 0 and $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ in radians that is coterminal with $\text{\hspace{0.17em}}-\frac{4\pi }{7}.$

Draw the angle $\text{\hspace{0.17em}}315°\text{\hspace{0.17em}}$ in standard position on the Cartesian plane.

Draw the angle $\text{\hspace{0.17em}}-\frac{\pi }{6}\text{\hspace{0.17em}}$ in standard position on the Cartesian plane.

A carnival has a Ferris wheel with a diameter of 80 feet. The time for the Ferris wheel to make one revolution is 75 seconds. What is the linear speed in feet per second of a point on the Ferris wheel? What is the angular speed in radians per second?

3.351 feet per second, $\text{\hspace{0.17em}}\frac{2\pi }{75}\text{\hspace{0.17em}}$ radians per second

Find the missing sides of the triangle $\text{\hspace{0.17em}}ABC:\mathrm{sin}\text{\hspace{0.17em}}B=\frac{3}{4},c=12.$

Find the missing sides of the triangle.

$a=\frac{9}{2},b=\frac{9\sqrt{3}}{2}$

The angle of elevation to the top of a building in Chicago is found to be 9 degrees from the ground at a distance of 2000 feet from the base of the building. Using this information, find the height of the building.

Find the exact value of $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\frac{\pi }{6}.$

$\frac{1}{2}$

Compute sine of $\text{\hspace{0.17em}}240°.$

State the domain of the sine and cosine functions.

real numbers

State the range of the sine and cosine functions.

Find the exact value of $\text{\hspace{0.17em}}\mathrm{cot}\text{\hspace{0.17em}}\frac{\pi }{4}.$

1

Find the exact value of $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\frac{\pi }{3}.$

Use reference angles to evaluate $\text{\hspace{0.17em}}\mathrm{csc}\text{\hspace{0.17em}}\frac{7\pi }{4}.$

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

Use reference angles to evaluate $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}210°.$

If $\text{\hspace{0.17em}}\text{csc}\text{\hspace{0.17em}}t=0.68,$ what is the $\text{\hspace{0.17em}}\text{csc}\left(-t\right)?$

–0.68

If $\text{\hspace{0.17em}}\text{cos}\text{\hspace{0.17em}}t=\frac{\sqrt{3}}{2},$ find $\text{\hspace{0.17em}}\text{cos}\left(t-2\pi \right).$

Find the missing angle: $\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\pi }{6}\right)=\mathrm{sin}\left(___\right)$

$\frac{\pi }{3}$

#### Questions & Answers

sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
Wrong question
Saad
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
f(x)= 1350. 2^(t/20); where t is in hours.
Merkeb

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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