<< Chapter < Page Chapter >> Page >

From the Pythagorean Theorem, we get

x 2 + y 2 = 1

Substituting x = 1 2 , we get

( 1 2 ) 2 + y 2 = 1

Solving for y , we get

1 4 + y 2 = 1         y 2 = 1 1 4         y 2 = 3 4           y = ± 3 2

Since t = π 3 has the terminal side in quadrant I where the y- coordinate is positive, we choose y = 3 2 , the positive value.

At t = π 3 (60°), the ( x , y ) coordinates for the point on a circle of radius 1 at an angle of 60° are ( 1 2 , 3 2 ) , so we can find the sine and cosine.

( x , y ) = ( 1 2 , 3 2 ) x = 1 2 , y = 3 2 cos t = 1 2 , sin t = 3 2

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 π 6 , or 30 π 4 , or 45° π 3 , or 60° π 2 , or 90°
Cosine 1 3 2 2 2 1 2 0
Sine 0 1 2 2 2 3 2 1

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

Graph of a quarter circle with angles of 0, 30, 45, 60, and 90 degrees inscribed. Equivalence of angles in radians shown. Points along circle are marked.

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 cos ( 30 ) 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 cos ( 5 π 3 ) using a graphing calculator or computer.

Enter the following keystrokes:

COS (   5   ×   π   ÷  3 ) ENTER

cos ( 5 π 3 ) = 0.5
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Evaluate sin ( π 3 ) .

approximately 0.866025403

Got questions? Get instant answers now!

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 0 and angles larger than 2 π can still be graphed on the unit circle and have real values of x , y , and 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 [ −1 , 1 ] . The bounds of the y -coordinate are also [ −1 , 1 ] . Therefore, the range of both the sine and cosine functions is [ −1 , 1 ] .

Graph of unit circle.

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.

Questions & Answers

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
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
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
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
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
Practice Key Terms 4

Get the best Precalculus course in your pocket!





Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask