Sum and Difference Formulas 1. sin(α + β)
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Name (Print):
Trigonometric Formulas
Time/Day of Class:
Sum and Difference Formulas 1. sin(α + β) = sin α cos β + sin β cos α 2. sin(α − β) = sin α cos β − sin β cos α 3. cos(α + β) = cos α cos β − sin α sin β 4. cos(α − β) = cos α cos β + sin α sin β 5. tan(α + β) =
tan α + tan β 1 − tan α tan β
6. tan(α − β) =
tan α − tan β 1 + tan α tan β
Half-Angle Formulas r α 1 − cos α 1. sin = ± 2 2 r 1 + cos α α 2. cos = ± 2 2 3. tan
Double-Angle Formulas 1. sin 2α = 2 sin α cos α 2. cos 2α = cos2 α − sin2 α = 2 cos2 α − 1 = 1 − 2 sin2 α 3. tan 2α =
2 tan α 1 − tan2 α
α 1 − cos α sin α = = 2 sin α 1 + cos α
Power-Reducing Formulas 1. sin2 α =
1 − cos 2α 2
2. cos2 α =
1 + cos 2α 2
Product-to-Sum Formulas 1 1. sin α sin β = [cos(α − β) − cos(α + β)] 2 1 2. cos α cos β = [cos(α − β) + cos(α + β)] 2 1 3. sin α cos β = [sin(α + β) + sin(α − β)] 2 1 4. cos α sin β = [sin(α + β) − sin(α − β)] 2
3. tan2 α =
1 − cos 2α 1 + cos 2α
1. Using the difference formula for cosines. (a) Use the unit circle to find cos(30◦ ) =
(b) Now use the difference formula for cosines. cos(30◦ ) = cos(90◦ − 60◦ ) =
2. Now calculate the sine for an angle we don’t already know. Use two reference angles from the unit circle to find sin(15◦ ) = sin(
◦
−
◦)
=
3. cos( 7π 12 ) =
4. Find the phase shift of sine and cosine. (a) Use the sum formula for sine to simplify sin(θ + π2 ) =
(b) Describe the relationship between sine and cosine in terms of horizontal shift.
5. Derive the double angle formulas using the sum formulas.
(a) cos(2θ) = cos(θ + θ) =
(b) sin(2θ) = sin(θ + θ) =
(c) tan(2θ) =
Page 2
6. If cos θ =
4 5
with θ in quadrant I, find sin(2θ).
(a) Use cos θ = lengths.
adjacent hypotenuse
to sketch a triangle that represents cos θ =
(b) Use your triangle and sin θ =
opposite hypotenuse
to determine sin θ.
sin(θ) =
(c) Use the double angle formula for sine to find sin 2θ.
7. Use the power reducing formula to simplify sin4 θ. sin4 θ = (sin2 θ)2 = 8. Derive the product to sum formula for sines. (a) Add the sum and difference formulas for sine. Use sin(α + β) = sin α cos β + sin β cos α and sin(α − β) = sin α cos β − sin β cos α to substitute into sin(α + β) + sin(α − β) =
(b) Simplify the right side.
(c) Divide both sides by 2 to get the product to sum formula for sine. 9. Simplify cos(3θ) sin(5θ).
Page 3
4 5.
Find any missing
Name (Print):
Trigonometric Formulas
Time/Day of Class:
Sum and Difference Formulas 1. sin(α + β) = sin α cos β + sin β cos α 2. sin(α − β) = sin α cos β − sin β cos α 3. cos(α + β) = cos α cos β − sin α sin β 4. cos(α − β) = cos α cos β + sin α sin β 5. tan(α + β) =
tan α + tan β 1 − tan α tan β
6. tan(α − β) =
tan α − tan β 1 + tan α tan β
Half-Angle Formulas r α 1 − cos α 1. sin = ± 2 2 r 1 + cos α α 2. cos = ± 2 2 3. tan
Double-Angle Formulas 1. sin 2α = 2 sin α cos α 2. cos 2α = cos2 α − sin2 α = 2 cos2 α − 1 = 1 − 2 sin2 α 3. tan 2α =
2 tan α 1 − tan2 α
α 1 − cos α sin α = = 2 sin α 1 + cos α
Power-Reducing Formulas 1. sin2 α =
1 − cos 2α 2
2. cos2 α =
1 + cos 2α 2
Product-to-Sum Formulas 1 1. sin α sin β = [cos(α − β) − cos(α + β)] 2 1 2. cos α cos β = [cos(α − β) + cos(α + β)] 2 1 3. sin α cos β = [sin(α + β) + sin(α − β)] 2 1 4. cos α sin β = [sin(α + β) − sin(α − β)] 2
3. tan2 α =
1 − cos 2α 1 + cos 2α
1. Using the difference formula for cosines. (a) Use the unit circle to find cos(30◦ ) =
(b) Now use the difference formula for cosines. cos(30◦ ) = cos(90◦ − 60◦ ) =
2. Now calculate the sine for an angle we don’t already know. Use two reference angles from the unit circle to find sin(15◦ ) = sin(
◦
−
◦)
=
3. cos( 7π 12 ) =
4. Find the phase shift of sine and cosine. (a) Use the sum formula for sine to simplify sin(θ + π2 ) =
(b) Describe the relationship between sine and cosine in terms of horizontal shift.
5. Derive the double angle formulas using the sum formulas.
(a) cos(2θ) = cos(θ + θ) =
(b) sin(2θ) = sin(θ + θ) =
(c) tan(2θ) =
Page 2
6. If cos θ =
4 5
with θ in quadrant I, find sin(2θ).
(a) Use cos θ = lengths.
adjacent hypotenuse
to sketch a triangle that represents cos θ =
(b) Use your triangle and sin θ =
opposite hypotenuse
to determine sin θ.
sin(θ) =
(c) Use the double angle formula for sine to find sin 2θ.
7. Use the power reducing formula to simplify sin4 θ. sin4 θ = (sin2 θ)2 = 8. Derive the product to sum formula for sines. (a) Add the sum and difference formulas for sine. Use sin(α + β) = sin α cos β + sin β cos α and sin(α − β) = sin α cos β − sin β cos α to substitute into sin(α + β) + sin(α − β) =
(b) Simplify the right side.
(c) Divide both sides by 2 to get the product to sum formula for sine. 9. Simplify cos(3θ) sin(5θ).
Page 3
4 5.
Find any missing
