5.3 Sum and Difference Identities HW: PG. Pg. 468-469 #1-20 e, 24

Cosine of a Difference Identity

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

Cosine of a Sum Cos(u + v) =

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

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

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

…Difference Sin(u - v)

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

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

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

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

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

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

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

Prove the identity: Sin2u = 2sinucosu

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

Prove the Identity: Cos4 - sin4 = cos2

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

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?

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

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

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

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

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

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

Deriving the Law of Sines

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

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

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

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

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

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.