The Harmonic Identities are useful when solving equations with the following forms.
A sin x ± B cos x = C or A cos x ± B sin x = C
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
In the examples that follow, I show you how using the harmonic identities helps solve these types of equations.
- 2sinθ – 3cosθ = 1 for 0° ≤ θ ≤ 360°
- 3cosθ – 3sinθ = 2 for 0° ≤ θ ≤ 360°