Chapter 5 Trigonometric Functions 5 4 Trigonometric Functions

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Chapter 5 Trigonometric Functions 5. 4 Trigonometric Functions of Real Numbers; Periodic Functions Copyright

Chapter 5 Trigonometric Functions 5. 4 Trigonometric Functions of Real Numbers; Periodic Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

Objectives: Use a unit circle to define trigonometric functions of real numbers. Recognize the

Objectives: Use a unit circle to define trigonometric functions of real numbers. Recognize the domain and range of sine and cosine functions. Use even and odd trigonometric functions. Use periodic properties. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2

The Unit Circle A unit circle is a circle of radius 1, with its

The Unit Circle A unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this circle is Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3

The Unit Circle (continued) In a unit circle, the radian measure of the central

The Unit Circle (continued) In a unit circle, the radian measure of the central angle is equal to the length of the intercepted arc. Both are given by the same real number t. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6

Definitions of the Trigonometric Functions in Terms of a Unit Circle Copyright © 2014,

Definitions of the Trigonometric Functions in Terms of a Unit Circle Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7

Example: Finding Values of the Trigonometric Functions Use the figure to find the values

Example: Finding Values of the Trigonometric Functions Use the figure to find the values of the trigonometric functions at t. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8

Example: Finding Values of the Trigonometric Functions Use the figure to find the values

Example: Finding Values of the Trigonometric Functions Use the figure to find the values of the trigonometric functions at t. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9

The Domain and Range of the Sine and Cosine Functions Copyright © 2014, 2010,

The Domain and Range of the Sine and Cosine Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10

Even and Odd Trigonometric Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11

Even and Odd Trigonometric Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11

Example: Using Even and Odd functions to Find Exact Values Find the exact value

Example: Using Even and Odd functions to Find Exact Values Find the exact value of each trigonometric function: a. b. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12

Definition of a Periodic Function Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13

Definition of a Periodic Function Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13

Periodic Properties of the Sine and Cosine Functions Copyright © 2014, 2010, 2007 Pearson

Periodic Properties of the Sine and Cosine Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14

Example: Using Periodic Properties to Find Exact Values Find the exact value of each

Example: Using Periodic Properties to Find Exact Values Find the exact value of each trigonometric function: a. b. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15

Periodic Properties of the Tangent and Cotangent Functions Copyright © 2014, 2010, 2007 Pearson

Periodic Properties of the Tangent and Cotangent Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16

Repetitive Behavior of the Sine, Cosine, and Tangent Functions Copyright © 2014, 2010, 2007

Repetitive Behavior of the Sine, Cosine, and Tangent Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17