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The Unit Circle:Finding Trig Functions At Any Angle

Table ofContents

Main MenuSelect any of the pages above or click the green button to the right to access the lesson’s table of contents.

QuizTest on knowledge of using the unit circle

HelpInstruction on how to navigate this module

CreditsContributors and links to resources used

BeginLesson

Table of ContentsSelect any of the topics from the listabove or click the green button to the right to begin learning.

Degrees & Radians Trig Functions

Quadrant I Angles

Other Angles

The Unit Circle

Summary

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Degrees/Radians

IntroductionThe unit circle is simply a

circle with a radius of 1, graphed on the Cartesian plane.

It provides a great way to find trig values of various angles.

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Table ofContents

(0,1)

(1,0)

(0,-1)

(-1,0)

r = 1 θ

TrigFunctions

Degrees & Radians

There are two ways to measure angles on the unit circle: by degrees and by radians. A full

revolution of a circle is 360°, and because the circumference of a circle is 2π r (and r =1), that length is also equivalent to 2π radians.

Given the total unit circle measurements, it can be divided now into smaller parts which will be

used to find various trig values.

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Table ofContents

0, 360°0, 2π

90°π/2

180°π

270°3π/2

45°

π/4

135°3π/4

225°5π/4

315°7π/4

Quadrant IAngles

Trig FunctionsBefore moving on to trig values on the unit circle, a

review of trig functions is necessary. The basic sine and cosine can be used to find all other trig functions.

Since the unit circle is on the Cartesian plane, it helps to know these functions then in terms of x- and y-coordinates.

Remember that if x or y is equal to zero, some of the trig values will be undefined.

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Sine

Cosine

Tangent =sine

cosine

Cotangent =cosinesine

Secant =

1cosine

Cosecant =1

sine= y

= x

=yx

=1x

=1y

=xy

x ≠ 0

x ≠ 0

y ≠ 0

y ≠ 0

OtherAngles

Quadrant I Angles

To find the x- and y-coordinates on the unit circle corresponding to the angles 30° (π/6), 45°

(π/4), and 60° (π/3), there are simple rules to follow:Each coordinate will have a 2 in the denominator. Then, the x-coordinates from the top will go, 1 – 2 – 3; and the y-coordinates will go, 3 – 2 – 1.

Remember to take the square root of each numerator.

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Table ofContents

(1,0)

(0,1)

45°

60°

30°

( , ) 2 2

( , ) 2 2

( , ) 2 2 1

√2

√3

1

√2

√3

The UnitCircle

Other AnglesThe coordinates in the other quadrants of the unit circle can be found in the same way.The x-coordinate (or cosine value) numerators

decrease when moving away from the x-axis, and the y-coordinate (or sine value) numerators increase when moving away from the x-axis.

The signs of each value may change however, depending on which quadrant the angle is in.

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Table ofContents

III

III IV

(0,1)

(1,0)

(0,-1)

(-1,0)

( + , + )

( + , - )

( - , + )

( - , - )

( , ) 2 2

( , ) 2 2

( , ) 2 2

-1

-√2

-√3

√2

√3

1

Summary

The Unit CircleWith that information, a complete unit circle can be formed, like shown above. Knowing that each ( x , y ) coordinate

corresponds to a (cosine , sine) value will help to find the values of any other trig function.

For an example, to find the trig function values at the angle 120° (or 2π/3), watch the animation above.

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Table ofContents( x , y ) = (cosine,

sine)

sin (120°) = √3/2

cos (120°) = -1/2

tan (120°) = sin/cos = (√3/2)/(-1/2) = -√3

cot (120°) = cos/sin = (-1/2)/(√3/2) = -1/√3

sec (120°) = 1/cos = 1/(-1/2) = -2csc (120°) = 1/sin = 1/(√3/2) = 2/√3

Quiz

Summary Video

Control the video with the buttons in the lower left corner. Closed captioning, annotation removal, and the option to watch the video on YouTube are all available via the buttons in the lower right corner.

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Table ofContents

Problem 1

QuizNow that you know how to use the unit circle, let’s test your knowledge with this short quiz!

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Problem #1Solve the following:

sin (3π/4) = x

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a) x = √2/2 b) x = -√2/2

c) x = 1 d) x = 135°

Choose and click on your answer below:

WrongSorry, your answer is incorrect. Click the green button on the rightto return to the problem, or click here to review the unit circle for help. Problem 1

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RightYour answer a) x = √2/2 is correct!Click the green button on the right to continue to problem #2.

Problem 2

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Problem #2Solve the following:

cos (4π/3) = x

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a) x = 1/2 b) x = -√3/2

c) x = 0 d) x = - 1/2

Choose and click on your answer below:

WrongSorry, your answer is incorrect. Click the green button on the rightto return to the problem, or click here to review the unit circle for help. Problem 2

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RightYour answer d) x = - 1/2 is correct!Click the green button on the right to continue to problem #3.

Problem 3

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Problem #3Solve the following:

tan (180°) = x

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a) x = DNE

b) x = 1

c) x = 0 d) x = - 1

Choose and click on your answer below:

WrongSorry, your answer is incorrect. Click the green button on the rightto return to the problem, or click here to review the unit circle for help. Problem 3

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RightYour answer c) x = 0 is correct!Click the green button on the right to continue to problem #4.

Problem 4

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Problem #4Solve the following:

sec (330°) = x

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a) x = 11π/6 b) x = DNE

c) x = 2/√3 d) x = √3/2

Choose and click on your answer below:

WrongSorry, your answer is incorrect. Click the green button on the rightto return to the problem, or click here to review the unit circle for help. Problem 4

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RightYour answer c) x = 2/√3 is correct!Click the green button on the right to continue to problem #5.

Problem 5

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Problem #5Solve the following:

cot (0) = cot (2π) = x

Main Menu Choose and click on

your answer below:

a) x = 1 b) x = DNE

c) x = 0 d) x = -1

WrongSorry, your answer is incorrect. Click the green button on the rightto return to the problem, or click here to review the unit circle for help. Problem 5

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RightYour answer b) x = DNE is correct!Click the green button on the right to continue to problem #6.

Problem 6

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Problem #6Solve the following:

csc (π/6) = x

Main Menu Choose and click on

your answer below:

a) x = 1/2 b) x = DNE

c) x = 2/√3 d) x = 2

WrongSorry, your answer is incorrect. Click the green button on the rightto return to the problem, or click here to review the unit circle for help. Problem 6

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RightYour answer b) x = 2 is correct!Click the green button on the right to complete the quiz.

Finish

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Quiz CompleteCongratulations! You have successfully completed the lesson and quiz over the unit circle!

Credits

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CreditsAbove are links to pictures and resources used for this module.

Author: Megan MooreModule Tester: Natasha Hinton

FinishLesson

Unit circle image retrieved from: http://etc.usf.edu/clipart/43200/43215/unit-circle7_43215_lg.gif

Unit circle information retrieved from: • http://www.mathsisfun.com/geome

try/unit-circle.html• http://coolmath.com/precalculus-re

view-calculus-intro/precalculus-trigonometry/28-the-unit-circle-01.htm

Unit circle Youtube video: patrickJMT

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