# 7.1 Angles  (Page 2/29)

 Page 2 / 29

Drawing an angle in standard position always starts the same way—draw the initial side along the positive x -axis. To place the terminal side of the angle, we must calculate the fraction of a full rotation the angle represents. We do that by dividing the angle measure in degrees by $\text{\hspace{0.17em}}360°.\text{\hspace{0.17em}}$ For example, to draw a $\text{\hspace{0.17em}}90°\text{\hspace{0.17em}}$ angle, we calculate that $\text{\hspace{0.17em}}\frac{90°}{360°}=\frac{1}{4}.\text{\hspace{0.17em}}$ So, the terminal side will be one-fourth of the way around the circle, moving counterclockwise from the positive x -axis. To draw a $\text{\hspace{0.17em}}360°$ angle, we calculate that $\text{\hspace{0.17em}}\frac{360°}{360°}=1.\text{\hspace{0.17em}}$ So the terminal side will be 1 complete rotation around the circle, moving counterclockwise from the positive x -axis. In this case, the initial side and the terminal side overlap. See [link] .

Since we define an angle in standard position    by its terminal side, we have a special type of angle whose terminal side lies on an axis, a quadrantal angle . This type of angle can have a measure of $\text{0°,}\text{\hspace{0.17em}}\text{90°,}\text{\hspace{0.17em}}\text{180°,}\text{\hspace{0.17em}}\text{270°,}$ or $\text{\hspace{0.17em}}\text{360°}.\text{\hspace{0.17em}}$ See [link] .

An angle is a quadrantal angle    if its terminal side lies on an axis, including $\text{0°,}\text{\hspace{0.17em}}\text{90°,}\text{\hspace{0.17em}}\text{180°,}\text{\hspace{0.17em}}\text{270°,}$ or $\text{\hspace{0.17em}}\text{360°}.$

Given an angle measure in degrees, draw the angle in standard position.

1. Express the angle measure as a fraction of $\text{\hspace{0.17em}}\text{360°}.$
2. Reduce the fraction to simplest form.
3. Draw an angle that contains that same fraction of the circle, beginning on the positive x -axis and moving counterclockwise for positive angles and clockwise for negative angles.

## Drawing an angle in standard position measured in degrees

1. Sketch an angle of $\text{\hspace{0.17em}}30°\text{\hspace{0.17em}}$ in standard position.
2. Sketch an angle of $\text{\hspace{0.17em}}-135°\text{\hspace{0.17em}}$ in standard position.
1. Divide the angle measure by $\text{\hspace{0.17em}}360°.$

$\frac{30°}{360°}=\frac{1}{12}$

To rewrite the fraction in a more familiar fraction, we can recognize that

$\frac{1}{12}=\frac{1}{3}\left(\frac{1}{4}\right)$

One-twelfth equals one-third of a quarter, so by dividing a quarter rotation into thirds, we can sketch a line at $\text{\hspace{0.17em}}30°,$ as in [link] .

2. Divide the angle measure by $\text{\hspace{0.17em}}360°.$

$\frac{-135°}{360°}=-\frac{3}{8}$

In this case, we can recognize that

$-\frac{3}{8}=-\frac{3}{2}\left(\frac{1}{4}\right)$

Negative three-eighths is one and one-half times a quarter, so we place a line by moving clockwise one full quarter and one-half of another quarter, as in [link] .

Show an angle of $\text{\hspace{0.17em}}240°\text{\hspace{0.17em}}$ on a circle in standard position. ## Converting between degrees and radians

Dividing a circle into 360 parts is an arbitrary choice, although it creates the familiar degree measurement. We may choose other ways to divide a circle. To find another unit, think of the process of drawing a circle. Imagine that you stop before the circle is completed. The portion that you drew is referred to as an arc. An arc may be a portion of a full circle, a full circle, or more than a full circle, represented by more than one full rotation. The length of the arc around an entire circle is called the circumference of that circle.

The circumference of a circle is $\text{\hspace{0.17em}}C=2\pi r.\text{\hspace{0.17em}}$ If we divide both sides of this equation by $\text{\hspace{0.17em}}r,$ we create the ratio of the circumference, which is always $\text{\hspace{0.17em}}2\pi ,$ to the radius, regardless of the length of the radius. So the circumference of any circle is $\text{\hspace{0.17em}}2\pi \approx 6.28\text{\hspace{0.17em}}$ times the length of the radius. That means that if we took a string as long as the radius and used it to measure consecutive lengths around the circumference, there would be room for six full string-lengths and a little more than a quarter of a seventh, as shown in [link] .

what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
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
I don't understand how radicals works pls
How look for the general solution of a trig function
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
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
cosA\1+sinA=secA-tanA
Wrong question
why two x + seven is equal to nineteen.
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.
simplify each radical by removing as many factors as possible (a) √75
how is infinity bidder from undefined? By By By OpenStax By Rhodes By JavaChamp Team By Brooke Delaney By OpenStax By Stephanie Redfern By OpenStax By Megan Earhart By OpenStax By Kevin Amaratunga