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Pre-CalculusChapter 4

Trigonometric Functions

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4.2 The Unit Circle

Objectives: Evaluate trigonometric functions using the unit circle.

Use domain and period to evaluate sine and cosine functions.

Use a calculator to evaluate trigonometric functions.

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What is the Unit Circle? Equation of the unit circle: x2 + y2 =

1 Center: (0, 0) Radius = 1

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Unit Circle with Number Line Imagine that the real number line is

wrapped around the unit circle, as shown.

Note: the positive numbers wrap towards the positive y-axis and the negative numbers wrap towards the negative y-axis.

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More Unit Circle Each real number t corresponds to a

point (x, y) on the circle.

Each real number t also corresponds to a central angle θ whose radian measure is t.

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Compare Values (8 Segments)

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Compare Values (12 Segments)

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Definition of Trig Functions Let t be a real number and let (x, y) be

the point on the unit circle corresponding to t. Then the six trig functions are defined:

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Example 1 Evaluate the six trig functions at

each real number.

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5.2

6.1

t

t

t

t

.4

3.3

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Exploration Complete the activity (handout) in

which you will investigate the periodic nature of the sine function as it relates to the unit circle. You will need a graphing calculator.

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Sine and Cosine

Domain:

Range: What happens

when we add 2π to t?

So, tt

tt

cos2cos

sin2sin

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In General For n revolutions around the unit

circle,

What is the period for sine and cosine?

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Example 2 Evaluate using its period

as an aid.

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13sin

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Even and Odd Functions Even Function if f (–t) = f

(t).

Odd Function if f (–t) = – f (t).

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Our Friend, the Calculator What do we need to always check

before solving a trig problem with a calculator?

We can easily solve for sine, cosine, or tangent. How do we solve for cosecant, secant, and cotangent?

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Homework 4.2 Worksheet 4.2