The R cos(θ ± α) Form

This lesson covers a powerful technique for rewriting expressions of the form a sin θ + b cos θ as a single trigonometric function R sin(θ + α) or R cos(θ + α). This technique — often called the "harmonic form" or "R-alpha method" — is essential for solving equations, finding maximum and minimum values, and modelling periodic phenomena.

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The General Result

Any expression of the form a sin θ + b cos θ can be written as:

a sin θ + b cos θ ≡ R sin(θ + α)

where R > 0 and 0 < α < 2π (or equivalently 0° < α < 360°).

Similarly, it can be written in the forms:

a sin θ + b cos θ ≡ R cos(θ − β)    (alternative form)
a cos θ + b sin θ ≡ R cos(θ − α)
a cos θ − b sin θ ≡ R cos(θ + α)

The choice of form depends on the question.

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Finding R and α

Method for a sin θ + b cos θ ≡ R sin(θ + α)