# 7.3 Unit circle  (Page 4/11)

 Page 4 / 11

From the Pythagorean Theorem, we get

${x}^{2}+{y}^{2}=1$

Substituting $\text{\hspace{0.17em}}x=\frac{1}{2},$ we get

${\left(\frac{1}{2}\right)}^{2}+{y}^{2}=1$

Solving for $\text{\hspace{0.17em}}y,$ we get

$\begin{array}{ccc}\hfill \frac{1}{4}+{y}^{2}& =& 1\hfill \\ \hfill {y}^{2}& =& 1-\frac{1}{4}\hfill \\ \hfill {y}^{2}& =& \frac{3}{4}\hfill \\ \hfill y& =& ±\frac{\sqrt{3}}{2}\hfill \end{array}$

Since $\text{\hspace{0.17em}}t=\frac{\pi }{3}\text{\hspace{0.17em}}$ has the terminal side in quadrant I where the y- coordinate is positive, we choose $\text{\hspace{0.17em}}y=\frac{\sqrt{3}}{2},$ the positive value.

At $\text{\hspace{0.17em}}t=\frac{\pi }{3}\text{\hspace{0.17em}}$ (60°), the $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ coordinates for the point on a circle of radius $\text{\hspace{0.17em}}1\text{\hspace{0.17em}}$ at an angle of $\text{\hspace{0.17em}}60°\text{\hspace{0.17em}}$ are $\text{\hspace{0.17em}}\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right),$ so we can find the sine and cosine.

We have now found the cosine and sine values for all of the most commonly encountered angles in the first quadrant of the unit circle. [link] summarizes these values.

 Angle $0$ $\frac{\pi }{6},$ or $\text{\hspace{0.17em}}30°$ $\frac{\pi }{4},$ or $\text{\hspace{0.17em}}45°$ $\frac{\pi }{3},$ or $\text{\hspace{0.17em}}60°$ $\frac{\pi }{2},$ or $\text{\hspace{0.17em}}90°$ Cosine 1 $\frac{\sqrt{3}}{2}$ $\frac{\sqrt{2}}{2}$ $\frac{1}{2}$ 0 Sine 0 $\frac{1}{2}$ $\frac{\sqrt{2}}{2}$ $\frac{\sqrt{3}}{2}$ 1

[link] shows the common angles in the first quadrant of the unit circle.

## Using a calculator to find sine and cosine

To find the cosine and sine of angles other than the special angles, we turn to a computer or calculator. Be aware : Most calculators can be set into “degree” or “radian” mode, which tells the calculator the units for the input value. When we evaluate $\text{\hspace{0.17em}}\mathrm{cos}\left(30\right)\text{\hspace{0.17em}}$ on our calculator, it will evaluate it as the cosine of 30 degrees if the calculator is in degree mode, or the cosine of 30 radians if the calculator is in radian mode.

Given an angle in radians, use a graphing calculator to find the cosine.

1. If the calculator has degree mode and radian mode, set it to radian mode.
2. Press the COS key.
3. Enter the radian value of the angle and press the close-parentheses key ")".
4. Press ENTER.

## Using a graphing calculator to find sine and cosine

Evaluate $\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{5\pi }{3}\right)\text{\hspace{0.17em}}$ using a graphing calculator or computer.

Enter the following keystrokes:

$\mathrm{cos}\left(\frac{5\pi }{3}\right)=0.5$

Evaluate $\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\pi }{3}\right).$

approximately 0.866025403

## Identifying the domain and range of sine and cosine functions

Now that we can find the sine and cosine of an angle, we need to discuss their domains and ranges. What are the domains of the sine and cosine functions? That is, what are the smallest and largest numbers that can be inputs of the functions? Because angles smaller than $\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and angles larger than $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ can still be graphed on the unit circle and have real values of $\text{\hspace{0.17em}}x,y,\text{and}\text{\hspace{0.17em}}r,$ there is no lower or upper limit to the angles that can be inputs to the sine and cosine functions. The input to the sine and cosine functions is the rotation from the positive x -axis, and that may be any real number.

What are the ranges of the sine and cosine functions? What are the least and greatest possible values for their output? We can see the answers by examining the unit circle, as shown in [link] . The bounds of the x -coordinate are $\text{\hspace{0.17em}}\left[-1,1\right].\text{\hspace{0.17em}}$ The bounds of the y -coordinate are also $\text{\hspace{0.17em}}\left[-1,1\right].\text{\hspace{0.17em}}$ Therefore, the range of both the sine and cosine functions is $\text{\hspace{0.17em}}\left[-1,1\right].$

## Finding reference angles

We have discussed finding the sine and cosine for angles in the first quadrant, but what if our angle is in another quadrant? For any given angle in the first quadrant, there is an angle in the second quadrant with the same sine value. Because the sine value is the y -coordinate on the unit circle, the other angle with the same sine will share the same y -value, but have the opposite x -value. Therefore, its cosine value will be the opposite of the first angle’s cosine value.

find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
factoring polynomial
find general solution of the Tanx=-1/root3,secx=2/root3
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
where can I get indices
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Need help with this question please
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
The sequence is {1,-1,1-1.....} has
how can we solve this problem
Sin(A+B) = sinBcosA+cosBsinA
Prove it
Eseka
Eseka
hi
Joel
yah
immy