1. Half-Angle Identities 11-5
Double-Angle and
Half-Angle Identities
11-5
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Algebra 2
Holt Algebra 2

2. Warm Up
Find tan θ for 0 ≤ θ ≤ 90°, if 1. 2. 3.

3. Objective
Evaluate and simplify expressions by using double-angle and half-angle identities.

4. You can use sum identities to derive the double-angle identities.
sin 2θ = sin(θ + θ)
= sinθ cosθ + cosθ sinθ
= 2 sinθ cosθ

5. You can derive the double-angle identities for cosine and tangent in the same way. There are three forms of the identity for cos 2θ, which are derived by using sin2θ + cos2θ = 1. It is common to rewrite expressions as functions of θ only.

6. Example 1: Evaluating Expressions with Double-Angle Identities
Find sin2θ and tan2θ if sinθ = and 0°<θ<90°.
Step 1 Find cosθ to evaluate sin2θ = 2sinθcosθ.
Method 1 Use the reference angle.
In Ql, 0° < θ < 90°, and sinθ =
x = 52
Use the Pythagorean
Theorem.
Solve for x. θ
r = 5
y = 2 x

7. Example 1 Continued
Method 2 Solve cos2θ = 1 – sin2θ.
cos2θ = 1 – sin2θ
cosθ =
Substitute for cosθ.
Simplify.

8. Example 1 Continued
Step 2 Find sin2θ.
sin2θ = 2sinθcosθ
Apply the identity for sin2θ.
Substitute for sinθ and
for cosθ.
Simplify.

9. Example 1 Continued
Step 3 Find tanθ to evaluate tan2θ =
Apply the tangent ratio identity.
Substitute for sinθ and
for cosθ.
Simplify.

10. Example 1 Continued
Step 4 Find tan 2θ.
Apply the identity for tan2θ.
Substitute for tan θ.

11. Example 1 Continued
Step 4 Continued
Simplify.

12. Caution!
The signs of x and y depend on the quadrant for angle θ.
sin cos Ql
Qll –
Qlll – –
QlV – +

13. Find tan2θ and cos2θ if cosθ = and 270°<θ<360°.
Check It Out! Example 1
Find tan2θ and cos2θ if cosθ = and 270°<θ<360°.
Step 1 Find tanθ to evaluate tan2θ =
Method 1 Use the reference angle.
In QlV, 270° < θ < 360°, and cosθ =
12 + y2 = 32
Use the Pythagorean
Theorem. θ
r=3
x=1
y= –2√ 2
Solve for y.

14. Check It Out! Example 1 Continued
Step 2 Find tan2θ.
tan2θ =
Apply the identity for tan2θ.
Substitute – for tanθ.
Simplify.

15. Check It Out! Example 1 Continued
Step 3 Find cos2θ.
cos2θ = 2cos2θ – 1
Apply the identity for cos2θ.
Substitute for cosθ.
Simplify.

16. You can use double-angle identities to prove trigonometric identities.

17. Example 2A: Proving identities with Double-Angle Identities
Prove each identity.
Choose the right-hand side to modify.
sin 2θ = 2tanθ – 2tanθ sin2θ
= 2tanθ (1– sin2θ)
Factor 2tanθ.
Rewrite using 1 –sin2θ = cos2θ.
= 2tanθ cos2θ
= 2(tanθcosθ)cosθ
Regroup.
Rewrite using tanθcosθ = sinθ.
= 2sinθcosθ
Apply the identity for sin2θ.
= sin2θ

18. Example 2B: Proving identities with Double-Angle Identities
cos2θ = (2 – sec2θ)(1 – sin2θ)
Choose the right-hand side to modify.
cos2θ = (2 – sec2θ)(1 – sin2θ)
Rewrite using 1 – sin2θ = cos2θ.
= (2 – sec2θ)(cos2θ)
= 2cos2θ – 1
Expand and simplify.
Apply the identity for cos2θ.
= cos2θ

19. Choose to modify either the left side or the right side of an identity
Choose to modify either the left side or the right side of an identity. Do not work on both sides at once.
Helpful Hint

20. Check It Out! Example 2a
Prove each identity.
cos4θ – sin4θ = cos2θ
(cos2θ – sin2θ)(cos2θ + sin2θ) =
Factor the left side.
(1)(cos2θ) =
Rewrite using
1 = cos2θ + sin2θ and cos2θ = cos2θ – sin2θ.
cos2θ = cos2θ
Simplify.

21. Check It Out! Example 2b
Prove each identity.
Rewrite tan θ ratio identity and Pythagorean identity.
Reciprocal sec θ identity and simplify fraction.

22. Check It Out! Example 2b Continued
Prove each identity.
Simplify.
Double angle identity.

23. You can use double-angle identities for cosine to derive the half-angle identities by substituting for θ. For example, cos2θ = 2 cos2θ – 1 can be rewritten as cosθ = 2 cos2 – 1. Then solve for cos

24. Half-angle identities are useful in calculating exact values for trigonometric expressions.

25. Example 3A: Evaluating Expressions with Half-Angle Identities
Use half-angle identities to find the exact value of cos 15°.
Positive in Ql.
Cos 30° =
Simplify.

26. Example 3A Continued
Check Use your calculator.

27. Example 3B: Evaluating Expressions with Half-Angle Identities
Use half-angle identities to find the exact value of
Negative in Qll.

28. Example 3B Continued
Simplify.

29. Example 3B Continued
Check Use your calculator.

30. Check It Out! Example 3a
Use half-angle identities to find the exact value of tan 75°.
tan (150°)
Positive in Ql.
Simplify.

31. Check It Out! Example 3a Continued
Check Use your calculator.

32. Check It Out! Example 3b
Use half-angle identities to find the exact value of
Negative in Qll.

33. Check It Out! Example 3b Continued
Simplify.
Check Use your calculator.

34. Example 4: Using the Pythagorean Theorem with Half-Angle Identities
Find cos and tan if tan θ = and 0<θ<
Step 1 Find cos θ to evaluate the half-angle identities. Use the reference angle.
In Ql, 0 < θ < and tanθ =
= x2
Pythagorean Theorem.
Solve for the missing side x.
Thus, cosθ =

35. Choose + for cos where 0 < θ <
Example 4 Continued
Step 2 Evaluate cos x 7 24 θ
Choose + for cos where 0 < θ <
Evaluate.

36. Example 4 Continued
Simplify.

37. Example 4 Continued
Step 3 Evaluate tan
Choose + for tan where
0 < θ <
Evaluate.

38. Example 4 Continued
Simplify.

39. Check It Out! Example 4
Find sin and cos if tan θ = and 0 < θ < 90.
Step 1 Find cos θ to evaluate the half-angle identities. Use the reference angle.
In Ql, 0 < θ < and tanθ =
= r 2
Pythagorean Theorem.
r =
Solve for the missing side r.
Thus, cosθ = .

40. Check It Out! Example 4 Continued
Step 2 Evaluate cos r 4 3 θ
Choose + for cos where 0 < θ <
Evaluate.
Simplify.

41. Check It Out! Example 4 Continued

42. Check It Out! Example 4 Continued
Step 3 Evaluate sin
Choose + for sin where 0 < θ < 90°.
Evaluate.

43. Check It Out! Example 4 Continued
Simplify.

44. Lesson Quiz: Part I
1. Find cos and cos 2θ if sin θ = and 0 < θ <
2. Prove the following identity:

45. Lesson Quiz: Part II
3. Find the exact value of cos 22.5°.