Download PC 4-2 The Unit Circle

Chapter 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.

1. t 
3. t  
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5
2. t 
4

3
4. t  
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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,
sin t  2   sin t
cost  2   cos t
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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
 13
Evaluate sin 
 6

 using its period as an aid.

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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
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