Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Trigonometric Functions of Any Angle Learn and use the definitions of the trigonometric functions of any angle. Learn and use the signs of the trigonometric functions. Learn to find and use a reference angle. Learn to find the area of an SAS triangle. Learn and use the unit circle definitions of the trigonometric functions. Learn and use some basic trigonometric identities. SECTION

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DEFINITION OF THE TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DEFINITION OF THE TRIGONOMETRIC FUNCTIONS OF ANY ANGLE  Let P(x, y) be any point on the terminal ray of an angle  in standard position (other than the origin), and let Then r > 0, and:

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1Finding Trigonometric Function Values Suppose that  is an angle whose terminal side contains the point P(–1, 3). Find the exact values of the six trigonometric functions of . Solution

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1Finding Trigonometric Function Values Solution continued

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley TRIGONOMETRIC FUNCTION VALUES OF COTERMINAL ANGLES These equations hold for any integer n.  in degrees  in radians

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley TRIGONOMETRIC FUNCTION VALUES OF QUADRANTAL ANGLES  deg  radians

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley SIGNS OF TRIGONOMETRIC FUNCTIONS

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DEFINITION OF A REFERENCE ANGLE Let  be an angle in standard position that is not a quadrantal angle. The reference angle for  is the positive acute angle  ´ (“theta prime”) formed by the terminal side of  and the positive or negative x-axis.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DEFINITION OF A REFERENCE ANGLE

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DEFINITION OF A REFERENCE ANGLE

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4Evaluating Trigonometric Functions Solution

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4Evaluating Trigonometric Functions Solution continued

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PROCEDURE FOR USING REFERENCE ANGLES TO FIND TRIGONOMETRIC FUNCTION VALUES Step 1If the degree measure of  is greater than 360º, then find a coterminal angle for  with degree measure between 0º and 360º. Otherwise, use  in Step 2. Step 2Find the reference angle  ´ for the angle resulting in Step 1. Write the trigonometric function of the acute angle,  ´.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PROCEDURE FOR USING REFERENCE ANGLES TO FIND TRIGONOMETRIC FUNCTION VALUES Step 3The sign of a trigonometric function of  depends on the quadrant in which  lies. Use the signs of the trigonometric functions to determine when to change the sign of the associated value for  ´. (Since  ´ is an acute angle, all its function values are positive.)

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Using the Reference Angle to Find Values of the Trigonometric Function Find the exact value of each expression. Solution Step 10º < 330º < 360º, find its reference angle Step 2330º is in Q IV, its reference angle  ´ is

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Using the Reference Angle to Find Values of the Trigonometric Function Solution continued Step 3In Q IV, tan is negative, so b. Step 1 is between 0 and 2π coterminal with

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Using the Reference Angle to Find Values of the Trigonometric Function Solution continued Step 3In Q IV, sec > 0, so Step 2 is in Q IV, its reference angle  ´ is

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley AREA OF A TRIANGLE In any triangle, if  is the included angle between sides b and c, the area K of the triangle is given by

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Finding a Triangular Area Determined by Cellular Telephone Towers Three cell towers are set up on three mountain peaks. Suppose the lines of sight from tower A to towers B and C form an angle of 120º, and the distances between tower A and towers B and C are 3.6 miles and 4.2 miles, respectively. Find the area of the triangle having these three towers as vertices.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Measuring the Height of Mount Kilimanjaro Solution Area of the triangle with angle  = 120º included between sides of lengths b = 3.6 and c = 4.2 is Finding a Triangular Area Determined by Cellular Telephone Towers

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley UNIT CIRCLE DEFINITIONS OF THE TRIGONOMETRIC FUNCTIONS OF REAL NUMBERS

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley UNIT CIRCLE DEFINITIONS OF THE TRIGONOMETRIC FUNCTIONS OF REAL NUMBERS Let t be any real number and let P(x, y) be the point on the unit circle associated with t. Then

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley BASIC TRIGONOMETRIC IDENTITIES Quotient Identities Reciprocal Identities Pythagorean Identities

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Finding the Exact Value of a Trigonometric Function Using the Pythagorean Identities a. Use Pythagorean identity involving sin t. Solution

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Finding the Exact Value of a Trigonometric Function Using the Pythagorean Identities Solution

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Finding the Exact Value of a Trigonometric Function Using the Pythagorean Identities Use Pythagorean identity involving sec t. Solution continued