<< 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

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
Practice Key Terms 3

Get the best Algebra and trigonometry course in your pocket!





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?

Ask