Slide 9- 1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Trigonometric Identities, Inverse Functions, and Equations Chapter 6

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.1 Identities: Pythagorean and Sum and Difference  State the Pythagorean identities.  Simplify and manipulate expressions containing trigonometric expressions.  Use the sum and difference identities to find function values.

Slide 9- 4 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Basic Identities An identity is an equation that is true for all possible replacements of the variables.

Slide 9- 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Pythagorean Identities

Slide 9- 6 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Multiply and simplify: a) Solution:

Slide 9- 7 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example continued b) Factor and simplify: Solution:

Slide 9- 8 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Another Example Simplify the following trigonometric expression: Solution:

Slide 9- 9 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sum and Difference Identities There are six identities here, half of them obtained by using the signs shown in color.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find sin 75  exactly.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.2 Identities: Cofunction, Double-Angle, and Half-Angle  Use cofunction identities to derive other identities.  Use the double-angle identities to find function values of twice an angle when one function value is known for that angle.  Use the half-angle identities to find function values of half an angle when one function value is known for that angle.  Simplify trigonometric expressions using the double- angle and half-angle identities.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Cofunction Identities Cofunction Identities for the Sine and Cosine

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find an identity for Solution:

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Double-Angle Identities

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find an equivalent expression for cos 3x. Solution:

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Half-Angle Identities

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find sin (  /8) exactly. Solution:

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Another Example Simplify. Solution:

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.3 Proving Trigonometric Identities  Prove identities using other identities.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Logic of Proving Identities Method 1: Start with either the left or the right side of the equation and obtain the other side. Method 2: Work with each side separately until you obtain the same expression.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Hints for Proving Identities Use method 1 or 2. Work with the more complex side first. Carry out any algebraic manipulations, such as adding, subtracting, multiplying, or factoring. Multiplying by 1 can be helpful when rational expressions are involved. Converting all expressions to sines and cosines is often helpful. Try something! Put your pencil to work and get involved. You will be amazed at how often this leads to success.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Prove the identity. Solution: Start with the left side.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Another Example Prove the identity: Solution: Start with the right side. Solution continued

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Prove the identity. Solution: Start with the left side.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.4 Inverses of the Trigonometric Functions  Find values of the inverse trigonometric functions.  Simplify expressions such as sin (sin –1 x) and sin –1 (sin x).  Simplify expressions involving compositions such as sin (cos –1 ) without using a calculator.  Simplify expressions such as sin arctan (a/b) by making a drawing and reading off appropriate ratios.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Inverse Trigonometric Functions [0,  ][  1, 1] RangeDomainFunction

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find each of the following: a) b) c) Solution: a) Find  such that  would represent a 60° or 120° angle.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution continued b) Find  such that  would represent a 30° reference angle in the 2 nd and 3 rd quadrants. Therefore,  = 150° or 210° c) Find  such that This means that the sine and cosine of  must be opposites. Therefore,  must be 135° and 315°.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Domains and Ranges

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Composition of Trigonometric Functions

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Examples Simplify: Since  1/2 is in the domain of sin –1, Simplify: Since is not in the domain of cos –1,

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Special Cases

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Examples Simplify: Since  /2 is in the range of sin –1, Simplify: Since  /3 is in the range of tan –1,

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley More Examples Simplify: Solution: Simplify: Solution:

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.5 Solving Trigonometric Equations  Solve trigonometric equations.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Trigonometric Equations Trigonometric Equation—an equation that contains a trigonometric expression with a variable. To solve a trigonometric equation, find all values of the variable that make the equation true.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solve 2 sin x  1 = 0. Solution: First, solve for sin x on the unit circle. The values  /6 and 5  /6 plus any multiple of 2  will satisfy the equation. Thus the solutions are where k is any integer.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graphical Solution We can use either the Intersect method or the Zero method to solve trigonometric equations. We graph the equations y 1 = 2 sin x  1 and y 2 = 0.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Another Example Solve 2 cos 2 x  1 = 0. Solution: First, solve for cos x on the unit circle.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graphical Solution Solve 2 cos 2 x  1 = 0. One graphical solution shown.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley One More Example Solve 2 cos x + sec x = 0 Solution: Since neither factor of the equation can equal zero, the equation has no solution.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graphical Solution 2 cos x + sec x

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Last Example Solve 2 sin 2 x + 3sin x + 1 = 0. Solution: First solve for sin x on the unit circle.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Last Example continued where k is any integer. One Graphical Solution