The R-Formula (Harmonic Form)

$a\cos\theta + b\sin\theta = R\cos(\theta - \alpha)$

where $R = \sqrt{a^2 + b^2}$ and $\tan\alpha = \frac{b}{a}$.

1. Expand $R\cos(\theta - \alpha) = R\cos\theta\cos\alpha + R\sin\theta\sin\alpha$.
2. Compare coefficients: $R\cos\alpha = a$, $R\sin\alpha = b$.
3. $R = \sqrt{a^2 + b^2}$, $\tan\alpha = \frac{b}{a}$.

$R\cos(\theta - \alpha)$ has max $R$ (when $\theta = \alpha$) and min $-R$ (when $\theta = \alpha + \pi$).

Write $3\cos\theta + 4\sin\theta$ in the form $R\cos(\theta - \alpha)$.

$R = \sqrt{9 + 16} = 5$. $\tan\alpha = \frac{4}{3}$ → $\alpha = 53.13°$.

$3\cos\theta + 4\sin\theta = 5\cos(\theta - 53.13°)$.

Max = 5, min = −5.

$3\cos\theta + 4\sin\theta = 2$ → $5\cos(\theta - 53.13°) = 2$ → $\cos(\theta - 53.13°) = 0.4$.

$\theta - 53.13° = \pm 66.42°$ → $\theta = 119.55°$ or $\theta = -13.29°$ (+ 360°).