Angle Addition Identities
You can determine the value of a trigonometric function of a given angle when you can express the sum (or difference) of angles that can comprise that angle.
sin(α + β) = sin α cos β + sin β cos α
sin(α − β) = sin α cos β − sin β cos α
cos(α + β) = cos α cos β − sin α sin β
cos(α − β) = cos α cos β + sin α sin β
sin(α − β) = sin α cos β − sin β cos α
cos(α + β) = cos α cos β − sin α sin β
cos(α − β) = cos α cos β + sin α sin β
tan(α + β) =
tan α + tan β
1 − tan α tan β
tan(α − β) = tan α − tan β 1 + tan α tan β
tan(α − β) = tan α − tan β 1 + tan α tan β
For example, sin 75 = sin (30 + 45) = sin 30 cos 45 + sin 45 cos 30
=
1
2
(
√2
2
)
+
(
√2
2
)
(
√3
2
)
= √2 4 + √6 4
= √2 + √6 4
= √2 4 + √6 4
= √2 + √6 4
Double Angle and Half Angle Identities
Double-angle identities are trig identities that can be used to rewrite trig functions that have a double angle.
Below are the double-angle identities and an example of how they are used.
sin(2θ) = 2 sin(θ) cos(θ)
cos(2θ) = cos²(θ) − sin²(θ)
= 1 − 2 sin²(θ)
= 2 cos²(θ) − 1
tan(2θ) = 2 tan θ 1 − tan²θ
cos(2θ) = cos²(θ) − sin²(θ)
= 1 − 2 sin²(θ)
= 2 cos²(θ) − 1
tan(2θ) = 2 tan θ 1 − tan²θ
Example: Given
sin(θ) =
4
5
and
0 < θ < π 2
Find sin (2θ):
0 < θ < π 2
Find sin (2θ):
sin(2θ) = 2
(
4
5
)
(
3
5
)
= 24 25
= 24 25
The half-angle formulas can be used in the same way.
Half-Angle Formulas
sin(θ/2) = ±
√
1 − cos θ
2
cos(θ/2) = ± √ 1 + cos θ 2
tan(θ/2) = ± 1 − cos θ sin θ = sin θ 1 + cos θ
cos(θ/2) = ± √ 1 + cos θ 2
tan(θ/2) = ± 1 − cos θ sin θ = sin θ 1 + cos θ
