Trigonometric Functions: The Unit Circle

MATH 109 - PrecalculusS. Rook

Overview

• Section 4.2 in the textbook:– Circular trigonometric functions– Properties of sine & cosine– Even & odd functions

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Circular Trigonometric Functions

Unit Circle

• Unit Circle: a special circle with a radius of 1, center of (0, 0), and equation of x2 + y2 = 1

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Real Number Line & Unit Circle

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• Starting at (1, 0) consider “wrapping” the positive real number line around the unit circle– Each number on the real number

line corresponds to ONE point on the unit circle

– Each point on the unit circle corresponds to MANY points on the real number line• Recall that the circumference of a circle is 2πr• Thus, each revolution around the unit circle constitutes

2π ≈ 6.28 units

Circular Functions

• Now consider a point (x, y) on the circumference of the unit circle where t is the length of the arc from (1, 0) to (x, y)

• Then – The central angle is equivalent

to the length of the arc it cuts on the Unit Circle

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tt

r

s1

Circular Functions (Continued)

• All points encountered on the unit circle can then be written as a function of t where t is the distance traveled from (1, 0)

t can be positive (counterclockwise) or negative (clockwise)

• There are six trigonometric functions – defined with respect to the Unit Circle they are:

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xt cosyt sin 0,tan xx

yt

0,1

sec xx

t0,1

csc yy

t 0,cot yy

xt

Circular Functions (Continued)• We call these the circular functions

– The radian measure of θ is the same as the arc length from (1, 0) to a point P on the terminal side of θ on the circumference of the unit circle

• Any point (x, y) on the unit circle can be written as (cos t, sin t)

• It is to your advantage to memorize the values in Quadrant I and the quadrantal angles (next slide)

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Common Angles and Function Values on the Unit Circle

Degrees Radians cos θ sin θ

0° 0 1 0

30°

45°

60°

90° 0 1

180° -1 0

270° 0 -1

360° 1 0

9

6

4

3

2

2

3

2

2

3

2

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2

2

1

2

2

2

1

2

1

2

1

Circular Functions (Example)

Ex 1: Use the Unit Circle to find the six trigonometric functions of:

a)

b)

10

4

7

3

4

Circular Functions (Example)

Ex 2: Use the Unit circle to find all values of t, 0 < t < 2π where

a)

b)

c) 11

2

1cos t

2

2sin t

1tan t

Circular Functions (Example)

Ex 3: If t is the positive distance from (1, 0) to point P along the circumference of the unit circle, sketch t on the circumference of the unit circle and then find the value of:

a) P = (0.8560, -0.5169); find i) cos t, ii) csc t, and iii) cot t

b) P = (0.0432, 0.9782); find i) sin t, ii) sec t, and iii) tan t

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Properties of Sine & Cosine

Domain & Range of Sine & Cosine

• Recall that domain is the allowable set of input values for a function:– Given (x, y) = (cos t, sin t) on the unit circle, there are no

places on the unit circle where cos t or sin t are undefined• Domain of f(t) = sin t and f(t) = cos t is (-oo, +oo)

• Recall that range is the acceptable output values for a function:– All points (x, y) = (cos t, sin t) on the unit circle must

satisfy: -1 ≤ (x, y) ≤ 1 • Range of f(t) = sin t and f(t) = cos t is [-1, 1]

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Period of Sine & Cosine

• We have already seen that ONE revolution around the unit circle occurs when t assumes values in the interval [0, 2π)

• Given a function f(t), the period is the smallest value c, c > 0 such that f(t + c) = f(t) for all t in the domain of f– i.e. when the function values start to repeat

• Given the point t = (x, y) on the unit circle, what value added to t results in the same point (x, y)? – Thus: sin(t + 2πn) = sin(t) and cos(t + 2πn) = cos(t), n is an

integer– What is the period of f(t) = sin t and f(t) = cos t

2π

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Period of Sine & Cosine (Example)

Ex 4: Evaluate the trigonometric function using its period [rewrite in the form cos(t + 2πn) or sin(t + 2πn)] and the unit circle:

a)

b)

16

3

8cos

2

15sin

Even and Odd Functions

Even and Odd Functions

• Recall the definition of even and odd functions:– If f(-t) = f(t), f(t) is an even function– If f(-t) = -f(t), f(t) is an odd function

• Examine the Unit Circle at the right:

cos(-θ) = cos θ meaning?sin(-θ) = -sin θ meaning?

Even Odd

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tt coscos

tt secsec

tt sinsin

tt csccsc

tt tantan tt cotcot

Even & Odd Functions (Example)

Ex 5: Use to evaluate:

a) cos t

b) sec(-t)

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5

1cos t

Summary

• After studying these slides, you should be able to:– Define the six trigonometric functions in terms of the unit

circle– Given a value of t or a point (x, y) evaluate the six

trigonometric functions– State the domain, range, and period of the sine and cosine– Understand even & odd functions

• Additional Practice– See the list of suggested problems for 4.2

• Next lesson– Right Triangle Trigonometry (Section 4.3)

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