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Trigonometric Functions: The Unit Circle 1.2

Objectives. Students will be able to identify a unit circle and describe its relationship to real numbers.Students will be able to use a unit circle to evaluate trigonometric functions.Students will be able to use the domain and period to evaluate sine and cosine functions.Students will be able t

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Trigonometric Functions: The Unit Circle 1.2

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    1. Trigonometric Functions: The Unit Circle 1.2

    2. Objectives Students will be able to identify a unit circle and describe its relationship to real numbers. Students will be able to use a unit circle to evaluate trigonometric functions. Students will be able to use the domain and period to evaluate sine and cosine functions. Students will be able to use a calculator to evaluate trigonometric functions.

    3. The Unit Circle

    4. Sine and Cosine and The Unit Circle

    5. Definitions of the Trigonometric Functions in Terms of a Unit Circle If t is a real number and (x, y) is a point on the unit circle that corresponds to t, then

    9. Examples Page 151 #6, 10, 16, 20, 26, 32 Try # 8, 14, 22, 28

    10. The Domain and Range of the Sine and Cosine Functions And Their Period The domain of the sine function and the cosine function is the set of all real numbers The range of these functions is the set of all real numbers from -1 to 1, inclusive. The period is 2p. This means it repeats every Periodic: f(t+c)=f(t) where c= 2p. Page 152 # 36, 42

    11. Definition of a Periodic Function A function f is periodic if there exists a positive number p such that f(t + p) = f(t) For all t in the domain of f. The smallest number p for which f is periodic is called the period of f.

    12. Periodic Properties of the Sine and Cosine Functions sin(t + 2?) = sin t and cos(t + 2?) = cos t The sine and cosine functions are periodic functions and have period 2?.

    13. Periodic Properties of the Tangent and Cotangent Functions tan(t + ?) = tan t and cot(t + ?) = cot t The tangent and cotangent functions are periodic functions and have period ?.

    14. Even and Odd Trigonometric Functions The cosine and secant functions are even. cos(-t) = cos t sec(-t) = sec t The sine, cosecant, tangent, and cotangent functions are odd. sin(-t) = -sin t csc(-t) = -csc t tan(-t) = -tan t cot(-t) = -cot t

    15. Example Use the value of the trigonometric function at t = ?/4 to find sin (- ?/4 ) and cos(- ?/4 ).

    16. Section 1.3 Right Angle Trigonometry

    17. Objectives Students will be able to evaluate trigonometric functions of acute angles. Students will be able to use fundamental trigonometric identities. Students will be able to use a calculator to evaluate trigonometric functions. Students will be able to use trigonometric functions to model and solve real life problems.

    18. The Six Trigonometric Functions

    19. Right Triangle Definitions of Trigonometric Functions How does compare to the unit circle? Page 160 #8, 12

    20. 45-45-90 Triangles 30-60-90 Triangles

    21. Reciprocal Identities

    22. Quotient Identities

    23. Pythagorean Identities

    24. Cofunction Identities

    25. Examples Page 161 #32, 38, 44, 46, 58, 62, 66, 70 Homework: 5 47 odd, 57 67 odd

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