Getting started
The following page uses trig identities,
and shows you how to solve slightly harder trig equations using the graphs. If you're just getting
started you might want to check out this page first.
This page uses degrees, but the methods are exactly the same if you
were to use radians. See this page
for help converting between degrees and radians.
Worked Example 1:
\[\begin{align}
\text{Find }\theta\hspace{6pt}\text{for } & 0\leq\theta\leq360
\\[8pt] \sin(\theta) &= \frac{1}{2}\cos(\theta)
\\[8pt] \frac{\sin(\theta)}{\cos(\theta)} &= \frac{1}{2}
\\[8pt] \tan(\theta) &= \frac{1}{2}
\\[8pt] \theta &= \tan^{-1}\left(\frac{1}{2}\right)
\\[8pt] &= 26.57\hspace{8pt}\text{[2dp]}
\\[10pt] \theta = 26.57^{\circ}\hspace{8pt} &\text{or}\hspace{8pt}206.57^{\circ}
\end{align}\]
Worked Example 2:
\[\begin{align}
\text{Find }\theta\hspace{6pt}\text{for } 0\leq\theta\leq360&
\\[8pt] \sin^3(\theta)+\sin(\theta)\cos^2(\theta)&=\frac{3}{4}
\\[8pt] \sin(\theta)(\sin^2(\theta)+\cos^2(\theta))&=\frac{3}{4}
\\[8pt] \sin(\theta)\times 1 &= \frac{3}{4}
\\[8pt] \text{[by }\sin^2(\theta)+\cos&^2(\theta)\equiv1\hspace{1pt}\text{]}&
\\[8pt] \theta = \sin^{-1}&\left(\frac{3}{4}\right)
\\[8pt] = 48.59&\hspace{8pt}\text{[2dp]}
\\[8pt] \theta = 48.59^{\circ}\hspace{8pt} \text{or}\hspace{8pt}&131.41^{\circ}
\end{align}\]
Worked Example 3:
\[\begin{align}
\text{Find }\theta\hspace{6pt}\text{for } & -180\leq\theta\leq180
\\[6pt] \sin^2(\theta) &= \frac{4}{\sec^2(\theta)}
\\[8pt] \sin^2(\theta) &= 4\cos^2(\theta)
\\[8pt] \frac{\sin^2(\theta)}{\cos^2(\theta)} &= 4
\\[8pt] \tan^2(\theta) &= 4
\\[8pt] \text{[by }\frac{\sin(\theta)}{\cos(\theta)}&\equiv\tan(\theta)\hspace{3pt}\text{]}
\\[8pt] \tan(\theta) &= \pm\sqrt{4} = \pm 2
\\[8pt] \theta &= \tan^{-1}(2)
\\[8pt] &= 63.43\hspace{8pt}\text{[2dp]}
\end{align}\]
You can use your calculator for \(\tan^{-1}(-2)=-63.43\) too but there's really no need when you use the graph properly and draw on both lines as above. \[\begin{align} \theta = &-116.57^{\circ}\text{, } -63.43^{\circ}\text{, } \\[6pt] &63.43^{\circ}\hspace{8pt}\text{or}\hspace{8pt} 116.57^{\circ} \end{align}\]
