LESSON 5 5 MultipleAngle and Product toSum Identities

LESSON 5– 5 Multiple-Angle and Product -to-Sum Identities

Five-Minute Check (over Lesson 5 -4) TEKS Then/Now Key Concept: Double-Angle Identities and Proof: Double-Angle Identity for Sine Example 1: Evaluate Expressions Involving Double Angles Example 2: Solve an Equation Using a Double-Angle Identity Key Concept: Power-Reducing Identities and Proof: Power-Reducing Identity for Sine Example 3: Use an Identity to Reduce a Power Example 4: Solve an Equation Using a Power-Reducing Identity Key Concept: Half-Angle Identities and Proof: Half-Angle Identity for Cosine Example 5: Evaluate an Expression Involving a Half Angle Example 6: Solve an Equation Using a Half-Angle Identity Key Concept: Product-to-Sum Identities and Proof: Product-to-Sum Identity for sin α cos β Example 7: Use an Identity to Write a Product as a Sum or Difference Key Concept: Sum-to-Product Identities and Proof: Sum-to-Product Identity for sin α + sin β Example 8: Use a Product-to-Sum or Sum-to-Product Identity Example 9: Solve an Equation Using a Sun-to-Product Identity

Over Lesson 5 -4 Find the exact value of sin 75°. A. B. C. D.

Over Lesson 5 -4 Find the exact value of A. B. C. D. .

Over Lesson 5 -4 Find the exact value of A. B. C. D. .

Over Lesson 5 -4 Simplify A. tan( + 19°) B. tan 19 C. tan(19 )° D. .

Over Lesson 5 -4 Find the solution to interval [0, 2 ). A. B. C. D. = 1 in the

Targeted TEKS P. 5(M) Use trigonometric identities such as reciprocal, quotient, Pythagorean, cofunctions, even/odd, and sum and difference identities for cosine and sine to simplify trigonometric Mathematical Processes P. 1(A), P. 1(D)

You used sum and difference identities. (Lesson 5 -4) • Use double-angle, power-reducing, and half-angle identities to evaluate trigonometric expressions and solve trigonometric equations. • Use product-to-sum identities to evaluate trigonometric expressions and solve trigonometric equations.

Evaluate Expressions Involving Double Angles If on the interval , find sin 2θ, cos 2θ, and tan 2θ. Since on the interval , one point on the terminal side of θ has y-coordinate 3 and a distance of 4 units from the origin, as shown.

Evaluate Expressions Involving Double Angles The x-coordinate of this point is therefore Using this point, we find that cos θ = tan θ = or and . Use these values and the double-angle identities for sine and cosine to find sin 2θ and cos 2θ. Then find tan 2θ using either the tangent double-angle identity or the definition of tangent. sin 2θ = 2 sin θ cos 2θ = 2 cos 2θ – 1 .

Evaluate Expressions Involving Double Angles Method 1

Evaluate Expressions Involving Double Angles Method 2

Evaluate Expressions Involving Double Angles Answer:

If on the interval cos 2 , and tan 2. A. B. C. D. , find, sin 2 ,

Solve an Equation Using a Double-Angle Identity Solve cos 2θ – cos θ = 2 on the interval [0, 2π]. cos 2θ – cos θ = 2 Original equation 2 cos 2 θ – 1 – cos θ – 2 = 0 Cosine Double-Angle Identity 2 cos 2 θ – cos θ – 3 = 0 Simplify. (2 cos θ – 3)( cos θ + 1) = 0 Factor. 2 cos θ – 3 = 0 or cos θ + 1 = 0 Zero Product Property cos θ = or cos θ = – 1 Solve for cos θ. θ = π Solve for

Solve an Equation Using a Double-Angle Identity Since cos θ = has no solution, the solution on the interval [0, 2π) is θ = π. Answer: π

Solve tan 2 + tan = 0 on the interval [0, 2π). A. B. C. D.

Use an Identity to Reduce a Power Rewrite csc 4 θ in terms of cosines of multiple angles with no power greater than 1. csc 4 θ = (csc 2 θ)2 = csc 4 θ Reciprocal Identity Pythagorean Identity Cosine Power. Reducing Identity

Use an Identity to Reduce a Power Common denominator Simplify. Square the fraction. Cosine Power. Reducing Identity

Use an Identity to Reduce a Power Common denominator Simplify. So, csc 4 θ = Answer: .

Rewrite tan 4 x in terms of cosines of multiple angles with no power greater than 1. A. B. C. D.

Solve an Equation Using a Power-Reducing Identity Solve sin 2 θ + cos 2θ – cos θ = 0. Solve Algebraically sin 2 θ + cos 2θ – cos θ = 0 Original equation Sine Power-Reducing Identity Multiply each side by 2. Add like terms. Double-Angle Identity Simplify.

Solve an Equation Using a Power-Reducing Identity Factor. 2 cos θ = 0 cos θ – 1 = 0 cos = 0 = cos = 1 =0 Zero Product Property Solve for cos . Solve for θ on [0, 2π). The graph of y = sin 2 θ + cos 2θ – cos θ has a period of 2 , so the solutions are

Solve an Equation Using a Power-Reducing Identity Support Graphically The graph of y = sin 2 θ + cos 2θ – cos θ has zeros at on the interval [0, 2π). Answer:

Solve cos 2 x + 2 cos 2 x = 0. A. B. C. D.

Evaluate an Expression Involving a Half Angle Find the exact value of sin 22. 5°. Notice that 22. 5° is half of 45°. Therefore, apply the half-angle identity for sine, noting that since 22. 5° lies in Quadrant I, its sine is positive. Sine Half-Angle Identity (Quadrant I angle)

Evaluate an Expression Involving a Half Angle Subtract and then divide. Quotient Property of Square Roots Answer:

Evaluate an Expression Involving a Half Angle CHECK Use a calculator to support your assertion that sin 22. 5° = and . sin 22. 5° = 0. 3826834324

Find the exact value of A. B. C. D. .

Solve an Equation Using a Half-Angle Identity Solve on the interval [0, 2π). Original equation Sine and Cosine Half. Angle Identities Square each side. 1 – cos x = 1 + cos x Multiply each side by 2. Subtract 1 – cos x from each side.

Solve an Equation Using a Half-Angle Identity Solve for cos x. Solve for x. The solutions on the interval [0, 2π) are Answer: .

Solve A. B. C. D. on the interval [0, 2π).

Use an Identity to Write a Product as a Sum or Difference Rewrite cos 6 x cos 3 x as a sum or difference. Product-to-Sum Identity Simplify. Distributive Property Answer:

Rewrite sin 4 x cos 2 x as a sum or difference. A. B. C. D.

Use a Product-to-Sum or Sum-to-Product Identity Find the exact value of cos 255° + cos 195°. Sum-to-Product Identity Simplify.

Use a Product-to-Sum or Sum-to-Product Identity Simplify. The exact value of cos 255° + cos 195° is Answer: .

Find the exact value of sin 255° + sin 195°. A. B. C. D.

Solve an Equation Using a Sum-to-Product Identity Solve sin 8 x – sin 2 x = 0. Solve Algebraically sin 8 x – sin 2 x = 0 Original equation Sine Sum-to-Product Identity Simplify. Set each factor equal to zero and find solutions on the interval [0, 2π).

Solve an Equation Using a Sum-to-Product Identity 2 cos 5 x = 0 First factor set equal to 0 Divide each side by 2. Multiple angle solutions in [0, 2π). Divide each solution by 5. sin 3 x = 0 Second factor set equal to 0 Multiple angle solutions in [0, 2π). Divide each solution by 3.

Solve an Equation Using a Sum-to-Product Identity The period of y = cos 5 x is y = sin 3 x is and the period of , so the solutions are where n is an integer.

Solve an Equation Using a Sum-to-Product Identity Support Graphically The graph of y = sin 8 x – sin 2 x has zeros at on the interval.

Solve an Equation Using a Sum-to-Product Identity Answer:

Solve sin 6 x + sin 2 x = 0. A. B. C. D.

LESSON 5– 5 Multiple-Angle and Product -to-Sum Identities