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.

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:

\[\begin{align} \text{Find }\theta\hspace{6pt}\text{for } & 0\leq\theta\leq360 \\[8pt] \sin(\theta) &= \frac{1}{2}\cos(\theta) \\[8pt] \frac{\sin(\theta)}{\cos(\theta)} &= \frac{1}{2} \\[8pt] \tan(\theta) &= \frac{1}{2} \\[8pt] \theta &= \tan^{-1}\left(\frac{1}{2}\right) \\[8pt] &= 26.57\hspace{8pt}\text{[2dp]} \\[10pt] \theta = 26.57^{\circ}\hspace{8pt} &\text{or}\hspace{8pt}206.57^{\circ} \end{align}\]
rough sketch of tan(t) to find solutions

Worked Example 2:

\[\begin{align} \text{Find }\theta\hspace{6pt}\text{for } 0\leq\theta\leq360& \\[8pt] \sin^3(\theta)+\sin(\theta)\cos^2(\theta)&=\frac{3}{4} \\[8pt] \sin(\theta)(\sin^2(\theta)+\cos^2(\theta))&=\frac{3}{4} \\[8pt] \sin(\theta)\times 1 &= \frac{3}{4} \\[8pt] \text{[by }\sin^2(\theta)+\cos&^2(\theta)\equiv1\hspace{1pt}\text{]}& \\[8pt] \theta = \sin^{-1}&\left(\frac{3}{4}\right) \\[8pt] = 48.59&\hspace{8pt}\text{[2dp]} \\[8pt] \theta = 48.59^{\circ}\hspace{8pt} \text{or}\hspace{8pt}&131.41^{\circ} \end{align}\]
rough sketch of sin(t) to find solutions

Worked Example 3:

\[\begin{align} \text{Find }\theta\hspace{6pt}\text{for } & -180\leq\theta\leq180 \\[6pt] \sin^2(\theta) &= \frac{4}{\sec^2(\theta)} \\[8pt] \sin^2(\theta) &= 4\cos^2(\theta) \\[8pt] \frac{\sin^2(\theta)}{\cos^2(\theta)} &= 4 \\[8pt] \tan^2(\theta) &= 4 \\[8pt] \text{[by }\frac{\sin(\theta)}{\cos(\theta)}&\equiv\tan(\theta)\hspace{3pt}\text{]} \\[8pt] \tan(\theta) &= \pm\sqrt{4} = \pm 2 \\[8pt] \theta &= \tan^{-1}(2) \\[8pt] &= 63.43\hspace{8pt}\text{[2dp]} \end{align}\]
rough sketch of tan(t) to find solutions
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. \[\begin{align} \theta = &-116.57^{\circ}\text{, } -63.43^{\circ}\text{, } \\[6pt] &63.43^{\circ}\hspace{8pt}\text{or}\hspace{8pt} 116.57^{\circ} \end{align}\]