<< Chapter < Page Chapter >> Page >

Find the coordinates of the point on the unit circle at an angle of 5 π 3 .

( 1 2 , 3 2 )

Got questions? Get instant answers now!

Key equations

Cosine cos t = x
Sine sin t = y
Pythagorean Identity cos 2 t + sin 2 t = 1

Key concepts

  • Finding the function values for the sine and cosine begins with drawing a unit circle, which is centered at the origin and has a radius of 1 unit.
  • Using the unit circle, the sine of an angle t equals the y -value of the endpoint on the unit circle of an arc of length t whereas the cosine of an angle t equals the x -value of the endpoint. See [link] .
  • The sine and cosine values are most directly determined when the corresponding point on the unit circle falls on an axis. See [link] .
  • When the sine or cosine is known, we can use the Pythagorean Identity to find the other. The Pythagorean Identity is also useful for determining the sines and cosines of special angles. See [link] .
  • Calculators and graphing software are helpful for finding sines and cosines if the proper procedure for entering information is known. See [link] .
  • The domain of the sine and cosine functions is all real numbers.
  • The range of both the sine and cosine functions is [ 1 , 1 ] .
  • The sine and cosine of an angle have the same absolute value as the sine and cosine of its reference angle.
  • The signs of the sine and cosine are determined from the x - and y -values in the quadrant of the original angle.
  • An angle’s reference angle is the size angle, t , formed by the terminal side of the angle t and the horizontal axis. See [link] .
  • Reference angles can be used to find the sine and cosine of the original angle. See [link] .
  • Reference angles can also be used to find the coordinates of a point on a circle. See [link] .

Section exercises

Verbal

Describe the unit circle.

The unit circle is a circle of radius 1 centered at the origin.

Got questions? Get instant answers now!

What do the x- and y- coordinates of the points on the unit circle represent?

Got questions? Get instant answers now!

Discuss the difference between a coterminal angle and a reference angle.

Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, t , formed by the terminal side of the angle t and the horizontal axis.

Got questions? Get instant answers now!

Explain how the cosine of an angle in the second quadrant differs from the cosine of its reference angle in the unit circle.

Got questions? Get instant answers now!

Explain how the sine of an angle in the second quadrant differs from the sine of its reference angle in the unit circle.

The sine values are equal.

Got questions? Get instant answers now!

Algebraic

For the following exercises, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by t lies.

sin ( t ) < 0 and cos ( t ) < 0

Got questions? Get instant answers now!

sin ( t ) > 0 and cos ( t ) > 0

I

Got questions? Get instant answers now!

sin ( t ) > 0 and cos ( t ) < 0

Got questions? Get instant answers now!

sin ( t ) < 0 and cos ( t ) > 0

IV

Got questions? Get instant answers now!

For the following exercises, find the exact value of each trigonometric function.

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 4

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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