Section 5.3: Double Angle, Power Reduction
and Half‐Angle Formulas
Each formula used in this section can be derived using the ‘grandparent’ formulas
from the previous section.
I. Double Angle Formulas
sin(2θ ) = 2sinθ cos θ
(only has one version)
cos(2θ ) = cos2 θ − sin2 θ
cos(2θ ) = 2cos2 θ − 1
cos(2θ ) = 1 − 2sin2 θ
(has THREE versions)
II. Power Reduction Formulas
(from cosine’s last two double angle formulas)
sin2 θ =
1 − cos(2θ )
2
cos2 θ =
1 + cos(2θ )
2
III. Half‐Angle Formulas
(from the power reduction formulas ... replace θ with α / 2 )
sin(α /2) = ±
1 − cos α
2
cos(α /2) = ±
1 + cos α
2
Now let’s put them to some use ...
I. Using the Double Angle Formulas
ex) Given that sinθ = 25 and θ acute, evaluate the following
b) cos2θ
a) sin2θ
ex) Verify the following identities:
a) (sinα + cos α) = 1 + sin2θ
2
sin2 2 x
= 2cos2 x
b)
1 − cos2 x
ex) Verify that sin4 θ = 4sin θ cos θ − 8sin3 θ cos θ
II. Using the Power Reduction Formulas
Their main purpose is to reduce the power of SQUARED sine and cosine
expressions.
ex) Rewrite the expression 16sin2 θ as a single powered cosine expression.
ex) Rewrite the expression 12sin2 β cos2 β as a single powered cosine expression.
III. Using the Half‐Angle Formulas*
*The “ + ” or “ – ” is determined by the half‐angle’s quadrant location
ex) Use the half‐angle formula to determine the exact value of :
a) sin(22.5°)
b) cos(105°)
ex) Given that cosθ = − 14 and θ lies in quadrant III, evaluate the following:
b) cos 2θ
a) sin 2θ