1. Concept

2. Find the value of cos 2 if sin = and  is between 0° and 90°.
Double-Angle Identities
Find the value of cos 2 if sin = and  is between 0° and 90°.
cos 2 = 1 – 2 sin2  Double-angle identity
Simplify.
Example 1

3. Step 1 Use the identity sin2 = 1 – cos2 to find the value of cos.
Double-Angle Identities
A. Find the exact value of tan 2 if cos = and  is between 0° and 90°.
Step 1 Use the identity sin2 = 1 – cos2 to find the value of cos.
cos2 + sin2 = 1
Subtract.
Example 2

4. Take the square root of each side.
Double-Angle Identities
Take the square root of each side.
Step 2 Find tan to use the double-angle identity for tan 2.
Definition of tangent
Example 2

5. Double-angle identity
Double-Angle Identities
Simplify.
Step 3 Find tan 2.
Double-angle identity
Example 2

6. Square the denominator and simplify.
Double-Angle Identities
Square the denominator and simplify.
Simplify.
Answer:
Example 2

7. Double-angle identity
Double-Angle Identities
B. Find the exact value of sin 2 if cos = and  is between 0° and 90°.
Double-angle identity
Simplify.
Answer:
Example 2

8. A. Find the exact value of cos2 if sin = and  is between 0° and 90°.
Example 2

9. B. Find the exact value of tan2 if sin = and  is between 0° and 90°.
Example 2

10. Concept

11. A. Find and  is the second quadrant.
Half-Angle Identities
A. Find and  is the second quadrant.
Since we must find cos first.
cos2 = 1 – sin2
sin2 + cos2 = 1
Simplify.
Example 3A

12. Take the square root of each side.
Half-Angle Identities
Take the square root of each side.
Since  is in the second quadrant,
Half-angle identity
Example 3

13. Rationalize the denominator.
Half-Angle Identities
Simplify the radicand.
Rationalize the denominator.
Multiply.
Example 3

14. Half-Angle Identities
Answer:
Example 3

15. B. Find the exact value of sin165.
Half-Angle Identities
B. Find the exact value of sin165.
165 is in Quadrant II; the value is positive.
Example 3B

16. Half-Angle Identities
Simplify.
Simplify.
Answer:
Example 3

17. A. Find and  is in the fourth quadrant.B. C. D.
Example 3A

18. B. Find the exact value of cos157.5.
Example 3B

19. Simplify Using Double-Angle Identities
FOUNTAIN Chicago’s Buckingham Fountain contains jets placed at specific angles that shoot water into the air to create arcs. When a stream of water shoots into the air with velocity v at an angle of  with the horizontal, the model predicts that the water will travel a horizontal distance of D = sin 2 and reach a maximum height of H = sin2. The ratio of H to D helps determine the total height and width of the fountain. Find
Example 4

20. Simplify the numerator and the denominator.
Simplify Using Double-Angle Identities
Original equation
Simplify the numerator and the denominator.
Example 4

21. Simplify. sin2 = 2sin cos Simplify.
Simplify Using Double-Angle Identities
Simplify.
sin2 = 2sin cos
Simplify.
Example 4

22. Simplify Using Double-Angle Identities
Answer:
Example 4

23. Verify that sin θ (cos2 θ – cos 2θ) = sin3 θ is an identity.
Verify Identities
Verify that sin θ (cos2 θ – cos 2θ) = sin3 θ is an identity.
Answer:
sin θ (cos2 θ – cos 2θ) = sin3 θ Original equation ?
sin θ [cos2 θ – (cos2 θ – sin2 θ)] = sin3 θ cos 2θ = cos2θ – sin2θ ?
sin θ (cos2 θ – cos2 θ + sin2 θ) = sin3 θ Distributive Property ?
sin θ (sin2 θ) = sin3 θ Simplify. ?
sin3 θ = sin3 θ Multiply.
Example 5

24. C. yes; cos2 θ + 2sin θ cos θ – sin2 θ = cos 2θ
Determine whether cos θ (cos θ + sin θ) – sin θ (cos θ + sin θ) = cos 2θ is an identity.
A. yes; cos2 θ – sin2 θ = cos 2θ
B. yes; cos2 θ – sin2 θ = 1
C. yes; cos2 θ + 2sin θ cos θ – sin2 θ = cos 2θ
D. no
Example 5