The document explains how to derive the compound angle formula for sin(A+B) using a diagram with two lines forming angles A and B. It shows that the angles marked (90-A) are equal, as are the angles equal to A. Using the relationships between trigonometric functions of equal angles, it rewrites sin(A+B) in terms of the sines and cosines of A and B, arriving at the formula sin(A+B)=BsinAcos+BcosAsin.

1. COMPOUND ANGLE FORMULAE: sin(A + B)
Draw a horizontal
base line.

2. Label this
angle A
A
Add a second line at
an angle to the first.
COMPOUND ANGLE FORMULAE: sin(A + B)

3. Label this
angle B
B
A
Add a third line
above the second.
COMPOUND ANGLE FORMULAE: sin(A + B)

4. B
A
T
S
R
Q
P
O
Call this point O
P !!
COMPOUND ANGLE FORMULAE: sin(A + B)
I’m off,
so keep up!!!

5. B
A
T
S
R
Q
P
O
COMPOUND ANGLE FORMULAE: sin(A + B)
Time to get
serious!!

6. B
A
T
S
R
Q
P
O
COMPOUND ANGLE FORMULAE: sin(A + B)
Can you identify all the equal angles?

7. B
A
T
S
R
Q
P
O
COMPOUND ANGLE FORMULAE: sin(A + B)
Those marked
equal (90 – A)o

8. B
A
T
S
R
Q
P
O
COMPOUND ANGLE FORMULAE: sin(A + B)
90 – A
These angles are
equal to angle A
A
A

9. COMPOUND ANGLE FORMULAE: sin(A + B)
B
A
T
S
R
Q
P
O
90 – A
A
B
A
T
S
R
Q
P
O
90 – A
A

10. COMPOUND ANGLE FORMULAE: sin(A + B)
B
A
T
S
R
Q
P
90 – A
A
sin(A + B)
OS
RS
=
OS
)TS+RT(
=
O

11. COMPOUND ANGLE FORMULAE: sin(A + B)
B
A
T
S
R
Q
P
90 – A
A
sin(A + B)
OS
RS
=
OS
)TS+RT(
=
Asin
OS
OQ
+Acos
OS
QS
=
AcosQS=TS
PQ=RT
AsinOQ=
QS
TS
=Acos
from this
we can rewrite this
as..
O

12. COMPOUND ANGLE FORMULAE: sin(A + B)
B
A
T
S
R
Q
P
O
90 – A
A
sin(A + B) Asin
OS
OQ
+Acos
OS
QS
=
Continuing….
Bsin=
OS
QS
Bcos=
OS
OQ
AsinBcos+AcosBsin=
BsinAcos+BcosAsin=
Rearrange…

13. COMPOUND ANGLE FORMULAE: sin(A + B)
In conclusion:
sin(A + B) BsinAcos+BcosAsin=
B
A

14. Did you get
that???