# Solving Trig Equations : Identities

## 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.

## 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}