Two elementary equations in sine and cosine

Statement

Solve the following equations with sine and cosine:

• (a) $2-\dfrac{1}{2}\sin(\alpha)=3+9\sin(\alpha)$
• (b) $1+3\cos(\theta)=-3\cos(\theta)-1$

Solution

(a) Collect the terms in $\sin\alpha$: $2-3=9\sin\alpha+\tfrac12\sin\alpha\Rightarrow -1=\tfrac{19}{2}\sin\alpha\Rightarrow \sin\alpha=-\frac{2}{19}\approx -0.105.$ Since $\arcsin(-\tfrac{2}{19})\approx -6.04°$, the solutions are $\alpha\approx -6.04°+360°k\quad\vee\quad \alpha\approx 186.04°+360°k,\qquad k\in\mathbb{Z}.$

(b) $1+1=-3\cos\theta-3\cos\theta\Rightarrow 2=-6\cos\theta\Rightarrow \cos\theta=-\frac13$. Hence $\theta\approx \pm 109.47°+360°k,\qquad k\in\mathbb{Z}.$

$\boxed{\sin\alpha=-\tfrac{2}{19}\ (\alpha\approx -6.04°\ \vee\ 186.04°);\quad \cos\theta=-\tfrac13\ (\theta\approx\pm 109.47°)}$