Compound angles

At the end of this chapter, you should be able to:

• state compound angle formulae for sin(A ± B),cos(A ± B) and tan(A ± B) • convert a sin ωt + b cos ωt into R sin(ωt +α) • derive double angle formulae • change products of sines and cosines into sums or differences • change sums or differences of sines and cosines into products • develop expressions for power in a.c. circuits – purely resistive, inductive and capacitive circuits, R-L and

R-C circuits

An electric current i may be expressed as i = 5sin(ωt −0.33) amperes. Similarly, the displacement x of a body from a fixed point can be expressed as x =10sin(2t +0.67)metres. The angles (ωt−0.33) and (2t +0.67) are called compound angles because they are the sum or difference of two angles. The compound angle formulae for sines and cosines of the sum and difference of two angles A and B are:

sin(A + B) = sin A cosB + cos A sin B sin(A − B) = sin A cosB − cos A sin B

cos(A + B) = cos A cosB − sin A sin B cos(A − B) = cos A cosB + sin A sin B

(Note, sin(A+B) is not equal to (sin A+ sin B), and so on.) The formulae stated above may be used to derive two further compound angle formulae:

tan(A + B) = tan A + tan B 1− tan A tan B

tan(A − B) = tan A − tan B 1+ tan A tan B

The for all values of A by A and B into the formulae they may be shown to be true.