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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
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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

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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.

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Find cosine and sine of the angle π .

cos ( π ) = 1 , sin ( π ) = 0

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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
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Questions & Answers

I don't understand how radicals works pls
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Himanshu Reply
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
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
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
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Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
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Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
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Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
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LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
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litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
Practice Key Terms 3

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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