1. 5.3 Sum and Difference Identities
HW: PG. Pg #1-20 e, 24

2. Cosine of a Difference Identity

4. Using the cosine-of-a-difference identity
Find the exact value of cos15° without using a calculator.

5. Cosine of a Sum
Cos(u + v) =

6. Cosine of a Sum or Difference
Cos (u v) = cosucosv sinusinv
The sign switches in either case**

7. Confirming cofunction Identities
Prove the identities:
Cos((π/2) - x) = sinx
Sin((π/2) - x) = cosx

8. Sine of a Difference or Sum
Sin(u + v) =

9. …Difference
Sin(u - v)

10. Sine of a Sum or Difference
Sin(u v) = sinucosv cosusinv
Signs do not switch in either case***

11. Using the sum/difference formulas
Write each as the sine or cosine of an angle:
Sin22ºcos13º + cos22ºsin13º
Cos(π/3)cos(π/4) + sin(π/3)sin(π/4)
Sinxsin2x - cosxcos2x

12. Proving Reduction Formulas
Sin(x + π) = -sinx
Cos(x + 3π/2) = sinx

13. Tangent of a Difference or Sum
Tan(u v) =

14. Prove the reduction formula: tan(θ - (3π/2)) = -cotθ

15. 5.4 - Multiple-Angle Identities
HW: Pg. 475 #12-40e

16. Double-Angle Identities
Sin2u = 2sinucosu
Cos2u = cos2u - sin2u
=2cos2u -1
=1 - 2sin2u
Tan2u = 2tanu/(1 - tan2u)

17. Prove the identity:
Sin2u = 2sinucosu

18. Power-Reducing Identities
Sin2u = (1-cos2u)/2
Cos2u = (1 + cos2u)/2
Tan2u = (1 - cos2u)/(1 +cos2u)

19. Prove the Identity:
Cos4 - sin4 = cos2

20. Reducing a power of 4
Rewrite cos4x in terms of trig functions with no power greater than 1.

21. Finding the Sine of Half an Angle (work with partner)
Recall sin2u = (1 - cos2u)/2
Use the power-reducing formula to show that sin2 (π/8) = (2 - √2)/4.
Solve for sin(π/8) . Do you take the positive or negative square root? Why?
Use the power-reducing formula to show that sin2 (9π/8) = (2 - √2)/4
Solve for sin (9π/8) . Do you take the positive or negative square root? Why?

22. Half-Angle Identities
Sin u/2 =
Cos u/2 =
Tanu/2 =

23. Solve algebraically in the interval [0,2π):
Sin2x = cosx
(use a double-angle identity)

24. Using half-angle Identities
Sin2x = 2sin2(x/2)

25. Practice with a Partner
Pg. 475 #1-39 odd

26. DO NOW: Find all solutions to the equation in the interval [0,2п)
Sin2x = 2sinx

27. 5.5 - The Law of Sines
HW: Pg. 484 #1-18e

28. Deriving the Law of Sines

29. Law of Sines
In any ∆ABC with angles A, B, and C opposite sides a, b, and c, then:
SinA / a = SinB / b = SinC / c

30. Solving a triangle given two angles and a side
Solve ∆ABC given that A = 36,
B = 48, and a = 8.
Solving Triangles (AAS, ASA)

31. Determining the Number of Triangles
EXPLORATION 1: Pg. 480 (work with partner) 1. 2. 3. 4. 5.

32. Solving a triangle given two sides and an angle
Solve ∆ABC given that a = 7, b = 6, and A = 26.3

33. Handling the ambiguous case
Solve ∆ABC given that a = 6, b = 7, and A = 30

34. Application Locating a Fire
Forest Ranger Daniel at ranger station A sights a fire in the direction 32 east of north. Ranger Preshan at ranger station B, 10 miles due east of A, sights the same fire on a line 48 west of north. Find the distance from each ranger station to the fire.