Copyright © 2008 Pearson Addison-Wesley. All rights reserved Fundamental Identities 5.2 Verifying Trigonometric Identities 5.3 Sum and Difference Identities for Cosine 5.4 Sum and Difference Identities for Sine and Tangent 5.5Double-Angle Identities 5.6Half-Angle Identities Trigonometric Identities 5

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-2 Trigonometric Identities 5.1 Fundamental Identities ▪ Using the Fundamental Identities

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-3 If and is in quadrant IV, find each function value. 5.1 Example 1 Finding Trigonometric Function Values Given One Value and the Quadrant (a) In quadrant IV, is negative.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-4 If and is in quadrant IV, find each function value. 5.1 Example 1 Finding Trigonometric Function Values Given One Value and the Quadrant (cont.) (b) (c)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 2 Expressing One Function in Terms of Another

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 3 Rewriting an Expression in Terms of Sine and Cosine Write in terms of and, and then simplify the expression.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-7 Verifying Trigonometric Identities 5.2 Verifying Identities by Working With One Side ▪ Verifying Identities by Working With Both Sides

Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-8 Verify that is an identity. 5.2 Example 1 Verifying an Identity (Working With One Side) Left side of given equation Right side of given equation

5-9 Verify that is an identity. 5.2 Example 2 Verifying an Identity (Working With One Side) Simplify.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Verify that is an identity. 5.2 Example 3 Verifying an Identity (Working With One Side) Factor. Simplify.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Verify that is an identity. 5.2 Example 4 Verifying an Identity (Working With One Side) Multiply by 1 in the form

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Verify that is an identity. 5.2 Example 5 Verifying an Identity (Working With Both Sides) Working with the left side: Multiply by 1 in the form Simplify.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 5 Verifying an Identity (Working With Both Sides) (cont.) Working with the right side: Factor the numerator. Distributive property. Factor the numerator. Factor the denominator. Simplify.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 5 Verifying an Identity (Working With Both Sides) (cont.) So, the identity is verified.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Sum and Difference Identities for Cosine 5.3 Difference Identity for Cosine ▪ Sum Identity for Cosine ▪ Cofunction Identities ▪ Applying the Sum and Difference Identities

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Find the exact value of each expression. 5.3 Example 1 Finding Exact Cosine Function Values (a)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 1 Finding Exact Cosine Function Values (cont.) (b) (c)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 2 Using Cofunction Identities to Find θ Find an angle θ that satisfies each of the following.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 3 Reducing cos ( A – B ) to a Function of a Single Variable Write cos(90° + θ) as a trigonometric function of θ alone.

Example 2B Using Cofunction Identities to Find θ ( Miscellaneous HW Examples ) Find an angle θ that satisfies each of the following. 1.Sin (θ + 15 o ) = Cos (2θ + 5 o ) Now see 5.3 # 38 2.Write Cos π/12 as cofunction Now see 5.3 #18

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 4 Finding cos ( s + t ) Given Information About s and t Suppose that, and both s and t are in quadrant IV. Find cos(s – t). The Pythagorean theorem gives Since s is in quadrant IV, y = –8.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 4 Finding cos ( s + t ) Given Information About s and t (cont.) Use a Pythagorean identity to find the value of cos t. Since t is in quadrant IV,

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 4 Finding cos ( s + t ) Given Information About s and t (cont.)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Sum and Difference Identities for Sine and Tangent 5.4 Sum and Difference Identities for Sine ▪ Sum and Difference Identities for Tangent ▪ Applying the Sum and Difference Identities

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Find the exact value of each expression. 5.4 Example 1 Finding Exact Sine and Tangent Function Values (a)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 1 Finding Exact Sine and Tangent Function Values (cont.) (b)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 1 Finding Exact Sine and Tangent Function Values (cont.) (c)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Write each function as an expression involving functions of θ. 5.4 Example 2 Writing Functions as Expressions Involving Functions of θ (a) (b) (c)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 4 Verifying an Identity Using Sum and Difference Identities Verify that the equation is an identity. Combine the fractions.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 4 Verifying an Identity Using Sum and Difference Identities (cont.) Expand the terms. Combine terms. Factor. So, the identity is verified.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Double-Angle Identities 5.5 Double-Angle Identities ▪ Omit Product-to-Sum and Sum-to- Product Identities

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 1 Finding Function Values of 2 θ Given Information About θ The identity for sin 2θ requires cos θ.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 1 Finding Function Values of 2 θ Given Information About θ (cont.)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 1 Finding Function Values of 2 θ Given Information About θ (cont.) Alternatively, find tan θ and then use the tangent double-angle identity.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 2 Finding Function Values of θ Given Information About 2 θ Find the values of the six trigonometric functions of θ if Use the identity to find sin θ: θ is in quadrant III, so sin θ is negative.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 2 Finding Function Values of θ Given Information About 2 θ (cont.) Use the identity to find cos θ: θ is in quadrant III, so cos θ is negative.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 2 Finding Function Values of θ Given Information About 2 θ (cont.)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Half-Angle Identities 5.6 Half-Angle Identities ▪ Applying the Half-Angle Identities

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 1 Using a Half-Angle Identity to Find an Exact Value Find the exact value of sin 22.5° using the half-angle identity for sine.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 2 Using a Half-Angle Identity to Find an Exact Value Find the exact value of tan 75° using the identity

Copyright © 2008 Pearson Addison-Wesley. All rights reserved The angle associated with lies in quadrant II since is positive while are negative. 5.6 Example 3 Finding Function Values of Given Information About s s2s2

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 3 Finding Function Values of Given Information About s (cont.) s2s2

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Inverse Circular Functions 6.2 Trigonometric Equations I 6.3 Trigonometric Equations II 6.4 Equations Involving Inverse Trigonometric Functions Inverse Circular Functions and Trigonometric Equations 6

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Inverse Circular Functions 6.1 Inverse Functions ▪ Inverse Sine Function ▪ Inverse Cosine Function ▪ Inverse Tangent Function ▪ Remaining Inverse Circular Functions ▪ Inverse Function Values

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Find y in each equation. 6.1 Example 1 Finding Inverse Sine Values

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 1 Finding Inverse Sine Values (cont.)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 1 Finding Inverse Sine Values (cont.) is not in the domain of the inverse sine function, [–1, 1], so does not exist. A graphing calculator will give an error message for this input.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Find y in each equation. 6.1 Example 2 Finding Inverse Cosine Values

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Find y in each equation. 6.1 Example 2 Finding Inverse Cosine Values

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 3 Finding Inverse Function Values (Degree- Measured Angles) Find the degree measure of θ in each of the following.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 4 Finding Inverse Function Values With a Calculator (a)Find y in radians if With the calculator in radian mode, enter as y =

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 4(b) Finding Inverse Function Values With a Calculator (b)Find θ in degrees if θ = arccot(–.2528). A calculator gives the inverse cotangent value of a negative number as a quadrant IV angle. The restriction on the range of arccotangent implies that the angle must be in quadrant II, so, with the calculator in degree mode, enter arccot(–.2528) as

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 4(b) Finding Inverse Function Values With a Calculator (cont.) θ = °

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 5 Finding Function Values Using Definitions of the Trigonometric Functions Evaluate each expression without a calculator. Since arcsin is defined only in quadrants I and IV, and is positive, θ is in quadrant I.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 5(a) Finding Function Values Using Definitions of the Trigonometric Functions (cont.)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 5(b) Finding Function Values Using Definitions of the Trigonometric Functions Since arccot is defined only in quadrants I and II, and is negative, θ is in quadrant II.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 5(b) Finding Function Values Using Definitions of the Trigonometric Functions (cont.)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 6(a) Finding Function Values Using Identities Evaluate the expression without a calculator. Use the cosine difference identity:

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 6(a) Finding Function Values Using Identities (cont.) Sketch both A and B in quadrant I. Use the Pythagorean theorem to find the missing side.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 6(a) Finding Function Values Using Identities (cont.)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 6(b) Finding Function Values Using Identities Evaluate the expression without a calculator. Use the double-angle sine identity: sin(2 arccot (–5)) Let A = arccot (–5), so cot A = –5.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 6(b) Finding Function Values Using Identities (cont.) Sketch A in quadrant II. Use the Pythagorean theorem to find the missing side.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 6(b) Finding Function Values Using Identities (cont.)

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Trigonometric Equations I 6.2 Solving by Linear Methods ▪ Solving by Factoring ▪ Solving by Quadratic Methods ▪ Solving by Using Trigonometric Identities

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 1 Solving a Trigonometric Equation by Linear Methods is positive in quadrants I and III. The reference angle is 30° because

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 1 Solving a Trigonometric Equation by Linear Methods (cont.) Solution set: {30°, 210°}

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 2 Solving a Trigonometric Equation by Factoring or Solution set: {90°, 135°, 270°, 315°}

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 3 Solving a Trigonometric Equation by Factoring

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 3 Solving a Trigonometric Equation by Factoring (cont.) has one solution, has two solutions, the angles in quadrants III and IV with the reference angle : and

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Trigonometric Equations II 6.3 Equations with Half-Angles ▪ Equations with Multiple Angles

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 1 Solving an Equation Using a Half-Angle Identity (a)over the interval and (b)give all solutions. is not in the requested domain.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 2 Solving an Equation With a Double Angle Factor. or

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 3 Solving an Equation Using a Multiple Angle Identity From the given interval 0 ° ≤ θ < 360°, the interval for 2θ is 0 ° ≤ 2θ < 720°.

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 3 Solving an Equation Using a Multiple Angle Identity (cont.) Since cosine is negative in quadrants II and III, solutions over this interval are

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Equations Involving Inverse Trigonometric Functions 6.4 Solving for x in Terms of y Using Inverse Functions ▪ Solving Inverse Trigonometric Equations

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 1 Solving an Equation for a Variable Using Inverse Notation

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 2 Solving an Equation Involving an Inverse Trigonometric Function

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 3 Solving an Equation Involving Inverse Trigonometric Functions

Copyright © 2008 Pearson Addison-Wesley. All rights reserved Example 3 Solving an Equation Involving Inverse Trigonometric Functions (cont.) Sketch u in quadrant I. Use the Pythagorean theorem to find the missing side.