The R-form (a sinθ + b cosθ)

Writing a sinθ ± b cosθ as a single R sin(θ ± α) or R cos(θ ∓ α), finding R and α by matching coefficients, and using the form to read off maxima/minima, solve equations and answer modelling questions.

Pure · Trigonometry

The R-form (a sinθ + b cosθ)

Why it works

A sum like

a\sin\theta + b\cos\theta

is really a single sine wave in disguise. Run an addition formula backwards: since

R\sin(\theta + \alpha) = R\cos\alpha\,\sin\theta + R\sin\alpha\,\cos\theta,

we can make this equal

a\sin\theta + b\cos\theta

by matching coefficients:

R\cos\alpha = a, \qquad R\sin\alpha = b.

Finding $R$ and $\alpha$ . Square and add:

R^2\cos^2\alpha + R^2\sin^2\alpha = a^2 + b^2

, so

R = \sqrt{a^2 + b^2}

(take

R > 0

). Divide:

\dfrac{R\sin\alpha}{R\cos\alpha} = \dfrac{b}{a}

, so

\tan\alpha = \dfrac{b}{a}

(take

\alpha

acute). The key discipline is to expand the chosen form, then match term by term — that tells you exactly which coefficient is

R\cos\alpha

and which is

R\sin\alpha

, so you never get the ratio upside down.

Use whichever target form keeps

\alpha

acute and the signs natural:

a\sin\theta + b\cos\theta = R\sin(\theta+\alpha), \qquad a\cos\theta + b\sin\theta = R\cos(\theta-\alpha).

What it's for.

Maximum and minimum. $R\sin(\cdots)$ swings between $-R$ and $R$ . So the

greatest value of

a\sin\theta + b\cos\theta

R = \sqrt{a^2+b^2}

(not

a+b

!), and the least is

-R

. The maximum occurs when the inside equals

90^\circ

(for the sine form), which pins down

\theta

Solving. Rewrite as $R\sin(\theta+\alpha) = c$ , so $\sin(\theta+\alpha) =

\tfrac{c}{R}

— a standard equation. **Transform the interval** to one for

\theta+\alpha

, solve, then subtract

\alpha$ back.

Modelling (height of tide, temperature, etc.): the model's amplitude is $R$

and its peak/trough come straight from the max/min above.