<< Chapter < Page Chapter >> Page >

Sine and cosine functions

If t is a real number and a point ( x , y ) on the unit circle corresponds to a central angle t , then

cos t = x
sin t = y

Given a point P ( x , y ) on the unit circle corresponding to an angle of t , find the sine and cosine.

  1. The sine of t is equal to the y -coordinate of point P : sin  t  =  y .
  2. The cosine of t is equal to the x -coordinate of point P : cos   t = x .

Finding function values for sine and cosine

Point P is a point on the unit circle corresponding to an angle of t , as shown in [link] . Find cos ( t ) and sin ( t ) .

Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (1/2, square root of 3 over 2).

We know that cos t is the x -coordinate of the corresponding point on the unit circle and sin t is the y -coordinate of the corresponding point on the unit circle. So:

x = cos t = 1 2 y = sin t = 3 2
Got questions? Get instant answers now!
Got questions? Get instant answers now!

A certain angle t corresponds to a point on the unit circle at ( 2 2 , 2 2 ) as shown in [link] . Find cos t and sin t .

Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (negative square root of 2 over 2, square root of 2 over 2).

cos ( t ) = 2 2 , sin ( t ) = 2 2

Got questions? Get instant answers now!

Finding sines and cosines of angles on an axis

For quadrantral angles, the corresponding point on the unit circle falls on the x- or y -axis. In that case, we can easily calculate cosine and sine from the values of x and y .

Calculating sines and cosines along an axis

Find cos ( 90° ) and sin ( 90° ) .

Moving 90° counterclockwise around the unit circle from the positive x -axis brings us to the top of the circle, where the ( x , y ) coordinates are ( 0 , 1 ) , as shown in [link] .

Graph of a circle with angle t, radius of 1, and a terminal side that intersects the circle at the point (0,1).

We can then use our definitions of cosine and sine.

x = cos  t = cos ( 90° ) = 0 y = sin  t = sin ( 90° ) = 1

The cosine of 90° is 0; the sine of 90° is 1.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find cosine and sine of the angle π .

cos ( π ) = 1 , sin ( π ) = 0

Got questions? Get instant answers now!

The pythagorean identity

Now that we can define sine and cosine, we will learn how they relate to each other and the unit circle. Recall that the equation for the unit circle is x 2 + y 2 = 1. Because x = cos t and y = sin t , we can substitute for x and y to get cos 2 t + sin 2 t = 1. This equation, cos 2 t + sin 2 t = 1 , is known as the Pythagorean Identity    . See [link] .

Graph of an angle t, with a point (x,y) on the unit circle. And equation showing the equivalence of 1, x^2 + y^2, and cos^2 t + sin^2 t.

We can use the Pythagorean Identity to find the cosine of an angle if we know the sine, or vice versa. However, because the equation yields two solutions, we need additional knowledge of the angle to choose the solution with the correct sign. If we know the quadrant where the angle is, we can easily choose the correct solution.

Pythagorean identity

The Pythagorean Identity    states that, for any real number t ,

cos 2 t + sin 2 t = 1

Given the sine of some angle t and its quadrant location, find the cosine of t .

  1. Substitute the known value of sin t into the Pythagorean Identity.
  2. Solve for cos t .
  3. Choose the solution with the appropriate sign for the x -values in the quadrant where t is located.

Finding a cosine from a sine or a sine from a cosine

If sin ( t ) = 3 7 and t is in the second quadrant, find cos ( t ) .

If we drop a vertical line from the point on the unit circle corresponding to t , we create a right triangle, from which we can see that the Pythagorean Identity is simply one case of the Pythagorean Theorem. See [link] .

Graph of a unit circle with an angle that intersects the circle at a point with the y-coordinate equal to 3/7.

Substituting the known value for sine into the Pythagorean Identity,

cos 2 ( t ) + sin 2 ( t ) = 1 cos 2 ( t ) + 9 49 = 1 cos 2 ( t ) = 40 49 cos ( t ) = ± 40 49 = ± 40 7 = ± 2 10 7

Because the angle is in the second quadrant, we know the x- value is a negative real number, so the cosine is also negative.

cos ( t ) = 2 10 7
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
Umesh Reply
I want to know trigonometry but I can't understand it anyone who can help
Siyabonga Reply
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
Sudip Reply
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
Sebit Reply
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
Marty Reply
I want to know partial fraction Decomposition.
Adama Reply
classes of function in mathematics
Yazidu Reply
divide y2_8y2+5y2/y2
Sumanth Reply
wish i knew calculus to understand what's going on 🙂
Dashawn Reply
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
Axmed Reply
24x^5
James
10x
Axmed
24X^5
Taieb
Thanks for this helpfull app
Axmed Reply
secA+tanA=2√5,sinA=?
richa Reply
tan2a+tan2a=√3
Rahulkumar
classes of function
Yazidu
if sinx°=sin@, then @ is - ?
NAVJIT Reply
the value of tan15°•tan20°•tan70°•tan75° -
NAVJIT
0.037 than find sin and tan?
Jon Reply
cos24/25 then find sin and tan
Deepak 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