<< 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

how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
Practice Key Terms 4

Get the best Precalculus course in your pocket!





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