1. Key Concept 1

2. Example 1 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.

3. Example 1 Evaluate Expressions Involving Double Angles 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 = 2sin θ cos θcos 2θ = 2cos 2 θ – 1 The x-coordinate of this point is therefore or. Using this point, we find that cos θ = and tan θ =.

4. Example 1 Evaluate Expressions Involving Double Angles Method 1

5. Example 1 Evaluate Expressions Involving Double Angles Method 2

6. Example 1 Evaluate Expressions Involving Double Angles Answer:

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

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

9. Key Concept 3

10. Example 3 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 2 θ) 2 = csc 4 θ Reciprocal Identity Pythagorean Identity Cosine Power- Reducing Identity

11. Example 3 Use an Identity to Reduce a Power Common denominator Simplify. Cosine Power- Reducing Identity Square the fraction.

12. Example 3 Use an Identity to Reduce a Power Simplify. Common denominator So, csc 4 θ =. Answer:

13. Example 4 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= 0Original equation Sine Power-Reducing Identity Add like terms. Double-Angle Identity Simplify. Multiply each side by 2.

14. Example 4 Solve an Equation Using a Power-Reducing Identity Factor. 2cos θ= 0cos θ – 1= 0Zero Product Property cos  = 0cos  = 1Solve for cos .  =  = 0Solve for θ on [0, 2 π ). The graph of y = sin 2 θ + cos 2θ – cos θ has a period of 2 , so the solutions are

15. Example 4 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, 2 π ). Answer:

16. Key Concept 5

17. Example 5 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)

18. Example 5 Evaluate an Expression Involving a Half Angle Quotient Property of Square Roots Subtract and then divide. Answer:

19. Example 5 Evaluate an Expression Involving a Half Angle CHECK Use a calculator to support your assertion that sin 22.5° =. sin 22.5° = 0.3826834324 and = 0.3826834324

20. Example 6 Solve an Equation Using a Half-Angle Identity Solve on the interval [0, 2π). Subtract 1 – cos x from each side. Square each side. Sine and Cosine Half- Angle Identities Original equation Multiply each side by 2.1 – cos x = 1 + cos x

21. Example 6 Solve an Equation Using a Half-Angle Identity Solve for cos x. Answer: Solve for x. The solutions on the interval [0, 2 π ) are.