# 5.1 Angles  (Page 4/29)

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## Using radians

Because radian    measure is the ratio of two lengths, it is a unitless measure. For example, in [link] , suppose the radius were 2 inches and the distance along the arc were also 2 inches. When we calculate the radian measure of the angle, the “inches” cancel, and we have a result without units. Therefore, it is not necessary to write the label “radians” after a radian measure, and if we see an angle that is not labeled with “degrees” or the degree symbol, we can assume that it is a radian measure.

Considering the most basic case, the unit circle (a circle with radius 1), we know that 1 rotation equals 360 degrees, 360°. We can also track one rotation around a circle by finding the circumference, $\text{\hspace{0.17em}}C=2\pi r,$ and for the unit circle $\text{\hspace{0.17em}}C=2\pi .\text{\hspace{0.17em}}$ These two different ways to rotate around a circle give us a way to convert from degrees to radians.

## Identifying special angles measured in radians

In addition to knowing the measurements in degrees and radians of a quarter revolution, a half revolution, and a full revolution, there are other frequently encountered angles in one revolution of a circle with which we should be familiar. It is common to encounter multiples of 30, 45, 60, and 90 degrees. These values are shown in [link] . Memorizing these angles will be very useful as we study the properties associated with angles.

Now, we can list the corresponding radian values for the common measures of a circle corresponding to those listed in [link] , which are shown in [link] . Be sure you can verify each of these measures.

## Finding a radian measure

Find the radian measure of one-third of a full rotation.

For any circle, the arc length along such a rotation would be one-third of the circumference. We know that

So,

$\begin{array}{l}\\ \begin{array}{l}s=\frac{1}{3}\left(2\pi r\right)\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{2\pi r}{3}\hfill \end{array}\end{array}$

The radian measure would be the arc length divided by the radius.

Find the radian measure of three-fourths of a full rotation.

$\frac{3\pi }{2}$

## Converting between radians and degrees

Because degrees and radians both measure angles, we need to be able to convert between them. We can easily do so using a proportion.

$\frac{\theta }{180}=\frac{{\theta }^{R}}{\pi }$

This proportion shows that the measure of angle $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ in degrees divided by 180 equals the measure of angle $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ in radians divided by $\text{\hspace{0.17em}}\pi .$ Or, phrased another way, degrees is to 180 as radians is to $\text{\hspace{0.17em}}\pi .$

$\frac{\text{Degrees}}{180}=\frac{\text{Radians}}{\pi }$

## Converting between radians and degrees

To convert between degrees and radians, use the proportion

$\frac{\theta }{180}=\frac{{\theta }^{R}}{\pi }$

## Converting radians to degrees

Convert each radian measure to degrees.

1. $\frac{\pi }{6}$
2. 3

Because we are given radians and we want degrees, we should set up a proportion and solve it.

1. We use the proportion, substituting the given information.
2. We use the proportion, substituting the given information.

#### Questions & Answers

what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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