Sum and Difference

1. Sin(3π-x) = sinx

1. Use the difference formula for Sine

sin(A-B) = (sinAcosB) - (cosAsinB)

2. Recognize the things you know

Sin3πcosx - cos3πsinx

3.Substitute in values

Sin3π = 0   cos3π = -1
(0)cosx - (-1)sinx = sinx

4. Simplify

0 + sinx = sinx
sinx = sinx

2. sin[(π/6)+x] = (1/2)(cosx + √3sinx)

1. use the sum formula for Sine

sin(A+B) = (sinAcosB) + (cosAsinB)
sin(π/6)cosx + cos(π/6)sinx

2. Recognize the things you know

sin(π/6)cosx + cos(π/6)sinx

3. Substitute in values

sin(π/6) = 1/2     cos(π/6) = (√3/2)
(1/2)cosx + (√3/2)sinx = (1/2)(cosx + √3sinx)

4. Factor out a half

(1/2)(cosx + √3sinx) = (1/2)(cosx + √3sinx)

3. sin(x+y) + sin(x-y) = 2sinxcosy

1. Use both the sum and the difference formula for Sine

sin(A-B) = (sinAcosB) - (cosAsinB)
sin(A+B) = (sinAcosB) + (cosAsinB)
(sinxcosy + sinycosx) + (sinxcosy - sinycosx) = 2sinxcosy

2. Combine like terms

sinycosx - sinycosx = 0     sinxcosy + sinxcosy = 2sinxcosy

2sinxcosy = 2sinxcosy

Double Angle

1. csc2Θ = (cscΘ) / (2cosΘ)

1. Get into terms of Sine and Cosine

1/sin2Θ = 1/(2sinΘcosΘ)


2. Recognize places to use identities
1/sin2Θ = 1/(2sinΘcosΘ)
3. Use the double angle formula for Sine

sin2Θ = 2sinΘcosΘ

1/(2sinΘcosΘ) = 1/(2sinΘcosΘ)

2. sec2Θ = (sec²Θ) / (2 - sec²Θ)


1. Get into terms of Cosine

1/cos2Θ = 1/ [cos²(2 - (1/cos²)]

2. Distribute 

1/cos2Θ = 1/ [cos²(2 - (1/cos²)]
1/cos2Θ = 1/ 2cos² - (cos²/cos²)                  (cos²/cos²) = 1
1/cos2Θ = 1/ 2cos² - 1

3. Use the double angle formula for Cosine

cos2Θ = 2cos² - 1

1/ 2cos² - 11/ 2cos² - 1

1/ 2cos² - 1 = 1/ 2cos² - 1


3. 1 + cos10y = 2cos²5y


1. Substitute Θ for 5y

    Θ = 5y
1 + cos2Θ = 2cos²Θ     

2. Use the double angle formula for Cosine
       
cos2Θ = 2cos² - 1

1 + 2cos² - 1 2cos²Θ 

2cos²Θ = 2cos²Θ3. Substitute 5y back in for Θ 
Θ = 5y


2cos²5y = 2cos²5y

Half Angle

1. sec(x/2) = +/- √[(2tanx)/(tanx + sinx)]

2. tan(x/2) = cscx - cotx


1. Use the half angle formula for Tangent

tan(x/2) = (1- cosx)/sinx
(1 - cosx)/sinx = cscx - cotx

2. Get into terms of Sine and Cosine

(1 - cosx)/sinx = (1/sinx) - (cosx/sinx)

3. Add Fractions

(1 - cosx)/sinx = (1/sinx) - (cosx/sinx)


(1 - cosx)/sinx = (1 - cosx)/sinx

3. -√[(1 + cos10x)/2] = -cos5x


1. Use the half angle formula for Cosine
Picture
 -cos(10x/2) = -cos5x
 10/2 = 5
-cos5x = -cos5x