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
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 - 1 = 1/ 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
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 - 1 = 1/ 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
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
-cos(10x/2) = -cos5x
10/2 = 5
-cos5x = -cos5x
10/2 = 5
-cos5x = -cos5x