The following identities, known as the harmonic identities, are very useful in solving certain types of trig. equation.
Try and learn them. However, it is a good exercise to try and prove them.
They are all very similar in the method of proof and are based on the addition identities.
I would encourage you to look at how I have proved one and then try the others as they are all very similar.
The Harmonic Identities:
- Asin(x) + Bcos(x) ≡ Rsin(x + α)
- Asin(x) – Bcos(x) ≡ Rsin(x – α)
- Acos(x) + Bsin(x) ≡ Rcos(x – α)
- Acos(x) – Bsin(x) ≡ Rcos(x + α)
where R = √ ( A2 + B2 ) and α = tan-1 B/A
Show:
- Asin(x) + Bcos(x) ≡ Rsin(x + α)
Show:
- Asin(x) – Bcos(x) ≡ Rsin(x – α)
Show:
- Acos(x) + Bsin(x) ≡ Rcos(x – α)
Show:
- Acos(x) – Bsin(x) ≡ Rcos(x + α)