1. 9-3: Other Identities

2. 9-3: Other Identities Double-Angle Identities Notes:
sin 2x = 2 sin x cos x
cos 2x = cos2 x – sin2 x
tan 2x =
Notes:
You never really need the tan 2x identity, because tan 2x = sin 2x/cos 2x
It’s also why we never bothered with the tan (x + y) identities yesterday

3. 9-3: Other Identities Ex 1: Use Double-Angle Identities
If and , find sin 2x and cos 2x
Since , we’re in the 3rd quadrant.
In the 3rd quadrant, sin is negative, cos is negative
Draw a triangle. -8 θ 17

4. 9-3: Other Identities Use Pythagorean Theorem to find the missing leg
b = -15 (remember: 3rd quadrant)
sin 2x = 2 sin x cos x
2 ● -15/17 ● -8/17 = 240/289
cos 2x = cos2 x – sin2 x
(-8/17)2 – (-15/17)2 = 64/289 – 225/289 = -161/289 -8 θ
-15 17

5. 9-3: Other Identities Example 2: Use Double-Angle Identities
Express f(x) = sin 3x in terms of powers of sin x and constants
sin 3x = sin (x + 2x)
= (sin x)(cos 2x ) + (sin 2x)(cos x)
= (sin x)(cos2 x – sin2 x) + (2 sin x cos x)(cos x)
= sin x cos2 x – sin3 x + 2 sin x cos2 x
= 3 sin x cos2 x – sin3 x
= 3 sin x (1 – sin2 x) – sin3 x
= 3 sin x – 3 sin3 x – sin3 x
= 3 sin x – 4 sin3 x

6. 9-3: Other Identities
Because cos 2x = cos2 x – sin2 x, we can use the Pythagorean Theorem to rewrite the identity of cos 2x to occasionally make solving/proving problems easier.
There are three forms of cos 2x
cos 2x = cos2 x – sin2 x
cos 2x = 1 – 2 sin2 x
cos 2x = 2 cos2 x – 1

7. 9-3: Other Identities Assignment Page 600 – 601
Problems 23 – 30 (all) and problem 43

8. 9-3: Other Identities
Power-Reducing Identities
sin2 x =
cos2 x =

9. 9-3: Other Identities
Example 4: Express f(x) = sin4 x in terms of constants and first powers of cosine functions
f(x) = sin4 x = sin2 x ● sin2 x
= ● = =

10. 9-3: Other Identities Half-Angle Identities
The sign in front of the radical depends upon the quadrant in which x/2 lies.

11. 9-3: Other Identities Example 5A: Find the exact value of cos
(since , x = )
(since cos is negative, so is cos x/2)

12. 9-3: Other Identities Example 5B: Find the exact value of sin
(since , x = )
(since sin is positive, so is sin x/2)

13. 9-3: Other Identities Alternate Half-Angle Identities for Tangent
The alternate identities remove the need to determine the sign
If and , find tan x/2

14. 9-3: Other Identities Ex 6: Use Half-Angle Identity for Tangent
Since , we’re in the 3rd quadrant.
In the 3rd quadrant, sin is negative, cos is negative
Draw a triangle. -2 θ -3

15. 9-3: Other Identities Use Pythagorean Theorem to find the missing leg
c =
sin x = cos x = -2 θ -3

16. 9-3: Other Identities Assignment Page 600 Show work
Problems 1 – 11 (odd)
Problems 31 – 35 (odd)
Show work

17. 9-3: Other Identities Product-to-Sum Identities
sin x cos y = ½ [sin(x + y) + sin(x – y)]
sin x sin y = ½ [cos(x – y) – cos(x + y)]
cos x cos y = ½ [cos(x + y) + cos(x – y)]
cos x sin y = ½ [sin(x + y) – sin(x – y)]
Sum-to-Product Identities
sin x + sin y = 2 sin cos
sin x – sin y = 2 cos sin
cos x + cos y = 2 cos cos
cos x – cos y = -2 sin sin

18. 9-3: Other Identities Ex 7: Use Sum-to-Product Identities
Prove the identity:
Numerator
Formula: sin x + sin y = 2 sin cos
sin t + sin 3t = 2 sin cos = 2 sin 2t cos (–t)
Denominator
Formula: cos x + cos y = 2 cos cos
cos t + cos 3t = 2 cos cos = 2 cos 2t cos (–t)
(Next slide)

19. 9-3: Other Identities

20. 9-3: Other Identities
Page 600
Problems 13 – 22 (all)