Elementary trigonometric equations (I)

Statement

Solve the following equations, drawing the unit circle:

• (a) $3-2\sin(\alpha)=1$
• (b) $3+2\sin(\alpha)=1$
• (c) $3+2\sin(\alpha)=1-2\sin(\alpha)$
• (d) $3+2\cos(\alpha)=1-2\cos(\alpha)$
• (e) $\cos(\alpha)=0$
• (f) $\cos(\alpha)=-\dfrac{\sqrt{3}}{2}$

Solution

Isolate the trigonometric function and read the angles on the unit circle (solutions in $[0,2\pi)$, plus the period $2k\pi$).

• (a) $\sin\alpha=1 \Rightarrow \alpha=\dfrac{\pi}{2}+2k\pi$.
• (b) $\sin\alpha=-1 \Rightarrow \alpha=\dfrac{3\pi}{2}+2k\pi$.
• (c) $4\sin\alpha=-2 \Rightarrow \sin\alpha=-\dfrac12 \Rightarrow \alpha=\dfrac{7\pi}{6}+2k\pi \ \vee\ \alpha=\dfrac{11\pi}{6}+2k\pi$.
• (d) $4\cos\alpha=-2 \Rightarrow \cos\alpha=-\dfrac12 \Rightarrow \alpha=\dfrac{2\pi}{3}+2k\pi \ \vee\ \alpha=\dfrac{4\pi}{3}+2k\pi$.
• (e) $\cos\alpha=0 \Rightarrow \alpha=\dfrac{\pi}{2}+k\pi$.
• (f) $\cos\alpha=-\dfrac{\sqrt3}{2} \Rightarrow \alpha=\dfrac{5\pi}{6}+2k\pi \ \vee\ \alpha=\dfrac{7\pi}{6}+2k\pi$.

$\boxed{\tfrac{\pi}{2};\ \tfrac{3\pi}{2};\ \tfrac{7\pi}{6},\tfrac{11\pi}{6};\ \tfrac{2\pi}{3},\tfrac{4\pi}{3};\ \tfrac{\pi}{2}+k\pi;\ \tfrac{5\pi}{6},\tfrac{7\pi}{6}}$