# 7.1 Angles  (Page 6/29)

 Page 6 / 29

Find an angle $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ that is coterminal with an angle measuring $\text{\hspace{0.17em}}870°,$ where $\text{\hspace{0.17em}}0°\le \alpha <360°.$

$\alpha =150°$

Given an angle with measure less than $\text{\hspace{0.17em}}0°,$ find a coterminal angle having a measure between $\text{\hspace{0.17em}}0°\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}360°.$

1. Add $\text{\hspace{0.17em}}360°\text{\hspace{0.17em}}$ to the given angle.
2. If the result is still less than $\text{\hspace{0.17em}}0°,$ add $\text{\hspace{0.17em}}360°\text{\hspace{0.17em}}$ again until the result is between $\text{\hspace{0.17em}}0°\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}360°.$
3. The resulting angle is coterminal with the original angle.

## Finding an angle coterminal with an angle measuring less than $\text{\hspace{0.17em}}0°$

Show the angle with measure $\text{\hspace{0.17em}}-45°\text{\hspace{0.17em}}$ on a circle and find a positive coterminal angle $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}0°\le \alpha <360°.$

Since $\text{\hspace{0.17em}}45°\text{\hspace{0.17em}}$ is half of $\text{\hspace{0.17em}}90°,$ we can start at the positive horizontal axis and measure clockwise half of a $\text{\hspace{0.17em}}90°\text{\hspace{0.17em}}$ angle.

Because we can find coterminal angles by adding or subtracting a full rotation of $\text{\hspace{0.17em}}360°,$ we can find a positive coterminal angle here by adding $\text{\hspace{0.17em}}360°.$

$-45°+360°=315°$

We can then show the angle on a circle, as in [link] .

Find an angle $\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}$ that is coterminal with an angle measuring $\text{\hspace{0.17em}}-300°\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}0°\le \beta <360°.$

$\beta =60°$

## Finding coterminal angles measured in radians

We can find coterminal angles    measured in radians in much the same way as we have found them using degrees. In both cases, we find coterminal angles by adding or subtracting one or more full rotations.

Given an angle greater than $\text{\hspace{0.17em}}2\pi ,$ find a coterminal angle between 0 and $\text{\hspace{0.17em}}2\pi .$

1. Subtract $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ from the given angle.
2. If the result is still greater than $\text{\hspace{0.17em}}2\pi ,$ subtract $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ again until the result is between $\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}2\pi .$
3. The resulting angle is coterminal with the original angle.

## Finding coterminal angles using radians

Find an angle $\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}$ that is coterminal with $\text{\hspace{0.17em}}\frac{19\pi }{4},$ where $\text{\hspace{0.17em}}0\le \beta <2\pi .$

When working in degrees, we found coterminal angles by adding or subtracting 360 degrees, a full rotation. Likewise, in radians, we can find coterminal angles by adding or subtracting full rotations of $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ radians:

$\begin{array}{ccc}\hfill \frac{19\pi }{4}-2\pi & =& \frac{19\pi }{4}-\frac{8\pi }{4}\hfill \\ & =& \frac{11\pi }{4}\hfill \end{array}$

The angle $\text{\hspace{0.17em}}\frac{11\pi }{4}\text{\hspace{0.17em}}$ is coterminal, but not less than $\text{\hspace{0.17em}}2\pi ,$ so we subtract another rotation.

$\begin{array}{ccc}\hfill \frac{11\pi }{4}-2\pi & =& \frac{11\pi }{4}-\frac{8\pi }{4}\hfill \\ & =& \frac{3\pi }{4}\hfill \end{array}$

The angle $\text{\hspace{0.17em}}\frac{3\pi }{4}\text{\hspace{0.17em}}$ is coterminal with $\text{\hspace{0.17em}}\frac{19\pi }{4},$ as shown in [link] .

Find an angle of measure $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ that is coterminal with an angle of measure $\text{\hspace{0.17em}}-\frac{17\pi }{6}\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}0\le \theta <2\pi .$

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

## Determining the length of an arc

Recall that the radian measure $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ of an angle was defined as the ratio of the arc length     $\text{\hspace{0.17em}}s\text{\hspace{0.17em}}$ of a circular arc to the radius $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ of the circle, $\text{\hspace{0.17em}}\theta =\frac{s}{r}.\text{\hspace{0.17em}}$ From this relationship, we can find arc length along a circle, given an angle.

## Arc length on a circle

In a circle of radius r , the length of an arc $\text{\hspace{0.17em}}s\text{\hspace{0.17em}}$ subtended by an angle with measure $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ in radians, shown in [link] , is

$s=r\theta$

Given a circle of radius $\text{\hspace{0.17em}}r,$ calculate the length $\text{\hspace{0.17em}}s\text{\hspace{0.17em}}$ of the arc subtended by a given angle of measure $\text{\hspace{0.17em}}\theta .$

1. If necessary, convert $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ to radians.
2. Multiply the radius $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta :s=r\theta .$

## Finding the length of an arc

Assume the orbit of Mercury around the sun is a perfect circle. Mercury is approximately 36 million miles from the sun.

1. In one Earth day, Mercury completes 0.0114 of its total revolution. How many miles does it travel in one day?
2. Use your answer from part (a) to determine the radian measure for Mercury’s movement in one Earth day.
1. Let’s begin by finding the circumference of Mercury’s orbit.

Since Mercury completes 0.0114 of its total revolution in one Earth day, we can now find the distance traveled.

2. Now, we convert to radians.

#### Questions & Answers

By the definition, is such that 0!=1.why?
Unikpel Reply
(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
Ajay Reply
hatdog
Mark
how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching
Shahid Reply
bsc F. y algebra and trigonometry pepper 2
Aditi Reply
given that x= 3/5 find sin 3x
Adamu Reply
4
DB
remove any signs and collect terms of -2(8a-3b-c)
Joeval Reply
-16a+6b+2c
Will
is that a real answer
Joeval
(x2-2x+8)-4(x2-3x+5)
Ayush Reply
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
2x²-6x+1=0
Ife
explain and give four example of hyperbolic function
Lukman Reply
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
Racelle Reply
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Anurag Reply
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
y2=4ax= y=4ax/2. y=2ax
akash
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
Jhovie Reply
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
Charmaine Reply
A banana.
Yaona
a function
Daniel
a function
emmanuel
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
Snalo Reply
what are you up to?
Mark Reply
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda

### Read also:

#### Get Jobilize Job Search Mobile App in your pocket Now!

Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

 By By By OpenStax By Mary Matera By Janet Forrester By OpenStax By OpenStax By By OpenStax By Jonathan Long By By OpenStax