Getting started
Trig equations often have multiple roots in the solution, here I go through some reasons why you might
"lose" some of them along the way! If you're just getting started with solving trig equations,
you might want to check out this page first.
For this page, I'm going to use degrees, but the methods are exactly the same if you
were to use radians. See this page
for help converting between degrees and radians.
Common Mistake 1: dividing by a \(\theta\) term
\[\begin{align} \text{Solve for}\hspace{8pt}0\leq\theta\leq& 360 \\[8pt] \sin(\theta)\cos(\theta)-\tfrac{1}{2}\sin(\theta)&=0 \\[8pt] \sin(\theta)\left(\cos(\theta)-\tfrac{1}{2}\right)&=0 \end{align}\] This next line is the important one, don't simply divide by \(\sin(\theta)\), you'd lose roots by cancelling! \[\sin(\theta)=0\hspace{8pt}\text{or}\hspace{8pt}\cos(\theta)-\tfrac{1}{2}=0\] Now solve separately for \(\theta\) to find all roots. \[\begin{align} \sin(\theta)&=0 \\[8pt] \theta=0^{\circ}, 180^{\circ}& \hspace{8pt}\text{or}\hspace{8pt}360^{\circ} \\[12pt] \cos(\theta)-\tfrac{1}{2}&=0 \\[8pt] \cos(\theta)&=\tfrac{1}{2} \\[8pt] \theta=60^{\circ}& \hspace{8pt}\text{or}\hspace{8pt}300^{\circ} \end{align}\]
\[\theta = 0^{\circ}, 60^{\circ}, 180^{\circ}, 300^{\circ}\hspace{8pt}\text{or}\hspace{8pt}360^{\circ}\] Be wary of this whenever you're faced with an opportunity to cancel a \(\theta\) term.
Common Mistake 2: square roots
\[\begin{align} \text{Solve for}\hspace{8pt}& 0\leq\theta\leq 360 \\[8pt] \cos^2(\theta)&=\tfrac{3}{4} \\[8pt] \cos(\theta)&=\pm\sqrt{\tfrac{3}{4}} \\[8pt] \cos(\theta)&=\pm\tfrac{\sqrt{3}}{2} \\[12pt]\cos(\theta)=\tfrac{\sqrt{3}}{2}\hspace{5pt}&\text{or}\hspace{5pt}\cos(\theta)=-\tfrac{\sqrt{3}}{2} \\[8pt] \cos(\theta)&=\tfrac{\sqrt{3}}{2} \\[8pt] \theta &= 30^{\circ}\hspace{8pt}\text{or}\hspace{8pt}330^{\circ} \\[8pt] \cos(\theta)&=-\tfrac{\sqrt{3}}{2} \\[8pt] \theta &= 150^{\circ}\hspace{8pt}\text{or}\hspace{8pt}210^{\circ} \end{align}\]
\[\theta=30^{\circ}, 150^{\circ}, 210^{\circ}\hspace{8pt}\text{or}\hspace{8pt}330^{\circ}\] The \(\pm\) is the really important bit here. Without it you'd lose half of your possible roots! We need to draw two lines for the two possibilities.
