5.5A – Compound Angle Formulas (Sum Identities)
Recall the special (common) angles and their exact ratios that can be generated from the special
triangles or the Unit circle.
xº
0º or 2π
x
30º or π/6
45º or π/4
60º or π/3
90º or π/2
π
π
π
π
6
4
1
3
2
3
2
1
2
1
2
0
1
3
-
0
Sin x
0
Cos x
1
Tan x
0
1
2
2
1
3
2
1
3
Using the Unit Circle one can recognize some basic relationships or identities.
Ex.
π
− x) = cos x and cos(
π
− x) = sin x
Co-function relationship
sin(
Supplemental relationship
sin θ o = sin(π − θ o ) and cos θ o = − cos(π − θ o )
Positional relationship
sin(− x) = − sin x and cos(− x) = cos x
2
2
So negative
angle will
give the same
ratio as
positive angle.
Just need to
adjust for sign
Complimentary
Further development using some basic geometry allows one to also come up with the following
formulas for adding and subtracting angles within a trigonometric function
Ex.
cos( x + y ) = cos x cos y − sin x sin y
cos( x − y ) = cos x cos y + sin x sin y
and
sin( x + y ) = sin x cos y + cos x sin y
sin( x − y ) = sin x cos y − cos x sin y
These compound (i.e. one angle made up from two other angles) formulas allows one to find
exact trigonometric ratios for angles other than the common ones listed in table above.
Ex.
a) 15º = 45º - 30º
2π π π
= +
b)
3
2 6
Example 1:
so
so
cos(15 o ) = cos 45 o cos 30 o + sin 45 o sin 30 o
⎛π
π⎞
π
π
π
Verify this formula
on your calculator by
comparing answers
to both sides.
π
sin ⎜ + ⎟ = sin cos + cos sin
2
6
2
6
⎝2 6⎠
Evaluate the following as exact answers
⎛π π ⎞
a) cos⎜ − ⎟ = cos π cos π + sin π sin π
⎝3 4⎠
3
4
3
4
1 1
3 1
= ⋅
+
⋅
2 2
2
2
Rationalize the
denominator for
final answer
5.5A – compound angle formula
=
=
1+ 3
2 2
2+ 6
4
⎛ 5π ⎞
b) sin ⎜
⎟
⎝ 12 ⎠
⎛ 5π ⎞
⎛π π ⎞
sin ⎜
⎟ = sin ⎜ + ⎟
⎝4 6⎠
⎝ 12 ⎠
= sin
Re-write
compound
angle as sum
or difference
of two known
exact ratios
=
=
=
1
2
π
4
⋅
cos
π
6
+ cos
3
1 1
+
⋅
2
2 2
3 +1
2 2
6+ 2
4
π
4
sin
π
6
5.5A –Compound Angle Formulas Practice Questions
1. Expand the following to express each as values of trigonometric functions of one number.
a) cos (x – 3)
b) sin (y – 4)
c) cos (2x + 3y)
d) sin (2x + 3y)
2. Use compound formula to express each of the following as a single trigonometric functions.
a) cos 2x cos x – sin 2x sin x
b) sin 5a cos 2a – cos 5a sin 2a
3. Given
4. Given
5. Given
6. Given
7. Given
b) sin 3x cos x + cos 3x sin x
d) cos x cos x – sin x sin x
5π π π
5π
= + Find cos
as exact value
12 4 6
12
π
12
=
π
3
−
π
4
Find cos
π
12
as exact value
2π π π
2π
= + Find sin
as exact value
3
2 6
3
π
12
=
π
3
−
π
4
Find sin
π
12
as exact value
2π π π
2π
= + Find cos
as exact value
3
2 6
3
8. Evaluate each of the following as an exact answer.
a) cos
5π
6
4π
3
5π
f) cos
4
b) sin
e) sin 15º
c) cos
3π
4
5π
4
5π
h) sin
6
d) sin
g) cos 75º
Answers 1. a) cos x cos 3 + sin x sin 3 b) sin y cos 4 – cos y sin 4 c) cos 2x cos 3y - sin 2x sin 3y
d) sin 2x cos 3y + cos 2x sin 3y 2. a) cos 3x b) sin 4x c) sin 3a d) cos 2x 3)
6)
3 −1
2 2
7) −
1
1
1
3
3
8. a) −
b) −
c) −
d) −
e)
2
2
2
2
2
5.5A – compound angle formula
6− 2
4)
4
1
6− 2
f) −
g)
4
2
6+ 2
3
5)
4
2
1
6− 2
h)
2
4