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Chapter 4 Trigonometric Functions 1
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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. 2
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What is the Unit Circle? Equation of the unit circle: x 2 + y 2 = 1 Center: (0, 0) Radius = 1 3
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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. 4
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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. 5
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Compare Values (8 Segments) 6
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Compare Values (12 Segments) 7
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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: 8
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Example 1 Evaluate the six trig functions at each real number. 9
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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. 10
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Sine and Cosine Domain: Range: What happens when we add 2π to t ? So, 11
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In General For n revolutions around the unit circle, What is the period for sine and cosine? 12
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Example 2 Evaluate using its period as an aid. 13
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Even and Odd Functions Even Function if f (–t) = f (t). Odd Function if f (–t) = – f (t). 14
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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? 15
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Homework 4.2 Worksheet 4.2 16
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