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Pre-Calculus

Pre-Calculus. Chapter 4 Trigonometric Functions. 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. What is the Unit Circle?.

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Pre-Calculus

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  1. Pre-Calculus Chapter 4 Trigonometric Functions

  2. 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.

  3. What is the Unit Circle? • Equation of the unit circle: x2 + y2 = 1 • Center: (0, 0) • Radius = 1

  4. 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.

  5. 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.

  6. Compare Values (8 Segments)

  7. Compare Values (12 Segments)

  8. 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:

  9. Example 1 • Evaluate the six trig functions at each real number.

  10. 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.

  11. Sine and Cosine • Domain: • Range: • What happens when we add 2πto t? • So,

  12. In General • For n revolutions around the unit circle, • What is the period for sine and cosine?

  13. Example 2 • Evaluate using its period as an aid.

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

  15. 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?

  16. Homework 4.2 • Worksheet 4.2

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