1. COMPOUND
ANGLE
FORMULAE

2. Consider the expression: sin (A + B).
Firstly note that sin (A + B) ≠ sin A + sin B
( This can easily be shown, e.g. let A = 30° and B = 60°).
It can be shown that the compound angle formula for sin (A + B) is:
. . . (1)
sin (A + B) = sinA cosB + cosA sinB
By letting B = – B in (1)
sin (A – B) = sinA cos(–B) + cosA sin(–B)
Since:
sin(–B) = – sinB
cos(–B) = cosB
. . . (2)
sin (A – B) = sinA cosB – cosA sinB

3. From (1):
sin (A + B) = sinA cosB + cosA sinB
Let A = 90 – A:
sin (90 – A + B) = sin(90 – A) cosB + cos(90 –A) sinB
sin (90 – (A – B)) = sin(90 – A) cosB + cos(90 –A) sinB
sin(90 – A) = cosA
cos(90 – A) = sinA
Since:
. . . (3)
cos (A – B) = cosA cosB + sinA sinB
Let B = – B in (3):
cos (A + B) = cosA cos(–B) + sinA sin(–B)
. . . (4)
cos (A + B) = cosA cosB – sinA sinB

4. sin (A + B) = sinA cosB + cosA sinB . . . (1)
cos (A + B) = cosA cosB – sinA sinB (4)
Divide (1) by (4):
tan (A + B) =
sinA cosB + cosA sinB
cosA cosB – sinA sinB
Divide each term by cosA cosB
tan (A + B) =
sinA cosB cosA sinB
cosA cosB sinA sinB
cosA cosB – +
. . . (5)
tan (A + B) =
tanA + tanB
1 – tanA tanB
. . . (6)
tan (A – B) =
tanA – tanB
1 + tanA tanB
Now let B = – B:

5. Example 1: Simplify sin(x + 90°).
sin (A + B) = sinA cosB + cosA sinB
Using:
sin (x + 90) =
sin x cos 90
+ cos x sin 90
= sin x ( 0 )
+ cos x ( 1 )
= cos x
Example 2: Find in surd form cos15°.
cos (A – B) = cosA cosB + sinA sinB
Using:
Let A = 45° and B = 30°
cos (45 – 30) = cos45 cos30 + sin45 sin30 = 2 2 + 2 1 2 4 = + 4 + 4 =

6. Example 3: Show that, in surd form tan75° = 2 +
tan (A + B) =
tanA + tanB
1 – tanA tanB
Using: = 1
1 +
1 – ×
tan ( ) =
tan45 + tan30
1 – tan45 tan30
Multiply each
term by: =
+ 1
– 1 ×
+ 1
– 1 =
(Rationalise here) =
3 + 2
+ 1
3 – 1 =
4 + 2 2
= 2 +

7. Example 4: Solve the equation sin (x + 30°) = 2 cos (x + 60°) ;
for 0 < x < 360°.
sin (x + 30°) = 2 cos (x + 60°)
sin x cos 30 + cos x sin 30
= 2{
cos x cos 60 – sin x sin 60 } 2 1 2 1 2 2 }
sin x +
cos x
cos x
= 2 {
sin x –
Multiply by 2:
sin x
cos x + =
2 cos x
sin x
– 2 = 3
sin x
cos x
Now, TanA is positive in
the 1st and 3rd quadrants: 3 1
tan x =
α = tan-1 3 1 
α = 10.89°
x = 10.9°, °
(1d.p.)

8. Example 5: Prove the identity cot (A + B) = cot A + cot B
tan (A + B) 1
tan (A + B) =
tanA + tanB
1 – tanA tanB
cot (A + B) =
LHS =
tan A + tan B
1 – tan A tan B =
Divide each term by tan A tan B
tan A tan B 1 =
– 1
tan B 1
tan A 1 +
cot B + cot A
cot A cot B – 1 =
= RHS

9. The compound angle formulae are:
Summary of key points:
The compound angle formulae are:
sin (A ± B) = sinA cosB ± cosA sinB
cos (A ± B) = cosA cosB sinA sinB
tan (A ± B) =
tanA ± tanB
1 tanA tanB
The following results also need to be known:
sin(–B) = – sinB
cos(–B) = cosB
sin(90 – A) = cosA
cos(90 – A) = sinA
This PowerPoint produced by R.Collins ; Updated Mar. 2010