The R-Formula (a sinθ ± b cosθ)

Any combination $a\sin\theta + b\cos\theta$ collapses into a single wave:

$a\sin\theta + b\cos\theta =$ $R\sin(\theta + \alpha)$, where $R = \sqrt{a^2 + b^2}$ and $\tan\alpha = \dfrac{b}{a}$

One function instead of two, which makes equations solvable and maxima readable. The R-formula question is among the most predictable in 0606, almost always in three parts.

Part 1. Express

Write $3\sin\theta + 4\cos\theta$ in the form $R\sin(\theta + \alpha)$, $0^\circ < \alpha < 90^\circ$. $R = \sqrt{9 + 16} =$ $5$ $\tan\alpha = \frac{4}{3}$ $\to$ $\alpha = 53.1^\circ$ $3\sin\theta + 4\cos\theta = 5\sin(\theta + 53.1^\circ)$

Derivation logic (worth knowing, occasionally demanded): expand $R\sin(\theta + \alpha) = R\cos\alpha\sin\theta + R\sin\alpha\cos\theta$ and match coefficients. $R\cos\alpha = a$, $R\sin\alpha = b$. Dividing gives $\tan\alpha = \dfrac{b}{a}$; squaring and adding gives $R^2 = a^2 + b^2$. The matching lines are method marks when “show that” appears.

Variants: $R\cos(\theta - \alpha)$ and the minus-sign forms each match differently, expand the target form and compare rather than memorising four cases. The question’s stated range for $\alpha$ tells you which quadrant it lives in; check your $\alpha$ lands there.

Part 2. Solve the equation

Hence solve $3\sin\theta + 4\cos\theta = 2$ for $0^\circ \le \theta \le 360^\circ$. $5\sin(\theta + 53.1^\circ) = 2$ $\to$ $\sin(\theta + 53.1^\circ) = 0.4$ $\theta + 53.1^\circ$ runs over $53.1^\circ$ to $413.1^\circ$, solve over that range: $\theta + 53.1^\circ = 156.4^\circ, 383.6^\circ$ $\theta = 103.3^\circ, 330.5^\circ$

The compound-angle range expansion is the standard equation discipline; the hence means the R-form is the intended route, solving from scratch scores poorly.

Part 3. Max, min, and where

The collapsed form answers instantly: maximum $R$ ($= 5$) where $\sin(\theta + \alpha) = 1$, i.e. $\theta = 90^\circ - \alpha = 36.9^\circ$; minimum $-R$ at $\theta = 270^\circ - \alpha$. Related: the maximum of $\dfrac{1}{3\sin\theta + 4\cos\theta + 7}$ occurs at the minimum of the denominator, a favourite twist.

Common mistakes

• $\tan\alpha = \frac{a}{b}$ (inverted)
• $\alpha$‘s quadrant unchecked against the stated range
• The equation solved without expanding the range for $\theta + \alpha$, solutions missing
• Max stated as $R +$ something when no constant exists (or the constant ignored when it does)
• “Hence” ignored in part 2

Full topic context: Trigonometry notes.