<< 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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 3

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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