### Archive

Posts Tagged ‘compound angle’

## Compound Angles: Cos(A+B) = CosACosB – SinASinB

Compound angles are angles made by adding two other angles together. When using trigonometry unfortunately you cant just “times out” the trig function but have to use an identity. This post will consider how we get the identity for cos(A+B):

Compound angle of A+B showing how they relate

From the definition of cos we find

cos(A+B) = OT/OR

but
OT = OP – PT
and PT = SQ so
OT = OP – SQ

so

cos(A+B) = ( OP – SQ ) / OR
so
cos(A+B) = OP/OR – SQ/OR

if we now times the both the top and bottom of the first term by OQ and do the same for the second term but with RQ we can get

but OP/OQ = cosB,
OQ/OR = cosA,
SQ/RQ = sinB,
RQ/OR = sinA

so we get, when these are substituted in and re arranged

cos(A+B) = cosAcosB-sinAsinB

Categories: maths, trigonometry

## Compound Angles – sin(A+B) = cosAsinB+sinAcosB

Compound angles are angles made by adding two other angles together. When using trigonometry unfortunately you cant just “times out” the trig function but have to use an identity. This post will consider how we get the identity for sin(A+B):

sin(A+B) = sinAcosB+sinBcosA

Compound angle of A+B showing how they relate

From the definition of sin=opp/hyp we find

sin(A+B) = RT/OR

But sinceRT comprises of RS+ST

By David Woodford

Categories: maths