Solving Trig Equations

Getting started

The following page uses inverses trig functions, and shows you how to solve simple trig equations using the graphs. An example question might be \[\begin{align} \text{Find }\theta\hspace{6pt}\text{for }&0\leq\theta\leq360 \\[6pt] \sin(\theta)&=0.5 \end{align}\] 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.

Worked Example: using the graphs

Let's take the previously mentioned question: \[\begin{align} \text{Find }\theta\hspace{6pt}\text{for }&0\leq\theta\leq360 \\[6pt] \sin(\theta)&=0.5 \end{align}\] We can simply use the inverse, \[\theta=\sin^{-1}(0.5)\] Here, you would use a calculator's \(\sin^{-1}\) function to get \[\theta = 30^\circ\] But this isn't the full answer! Let's look at the graph for \(\sin(\theta)=0.5\):

Having drawn the graph for \(\sin(\theta)=0.5\), we can see that there are multiple solutions for \(\theta\). Our calculator has only given us one value, so what can we do to find any others?

This process might seem long when explained but practice a few times and you'll find it really is pretty quick! (See 2nd example)

Let's focus on the range we're given: \(0\leq\theta\leq360\) - this region is shaded pink. Now there's only one solution we don't know. Because of the symmetry in a sine curve, and our knowledge that it has a zero at \(\theta=180\), we can work out that the other solution is \(\theta=180-30=150\) as shown below.

Note: you don't need to be exact with your curve sketch, you're not reading values off of it, just using it to help visualise where the solutions lie! We can conclude that \[\theta=30^\circ\hspace{8pt}\text{or }150^\circ\]

Worked Example: easy once you know how (1)

This is what it would look like to solve a similar problem with a sketch.

\[\begin{align} \text{Find }\theta\hspace{6pt}\text{for }&-180\leq\theta\leq180 \\[6pt] \cos(\theta)&=0.7 \\[10pt] \theta&=\cos^{-1}(0.7) \\[6pt] &= 45.57\hspace{8pt}\text{[2dp]} \end{align}\] This first value comes from using your calculator, use the sketch for other solutions in the range. \[\theta=-45.57^{\circ}\hspace{8pt}\text{or}\hspace{8pt}45.57^{\circ}\]

Worked Example: easy once you know how (2)

\[\begin{align} \text{Find }\theta\hspace{6pt}\text{for }&0\leq\theta\leq180 \\[6pt] \tan(\theta)&=-2.5 \\[10pt] \theta&=\tan^{-1}(-2.5) \\[6pt] &= -68.20\hspace{8pt}\text{[2dp]} \end{align}\] This first value comes from using your calculator, it's outside the range we're looking at (see here for information on principal values) but we still use the sketch in the same way. \[\theta=111.80^{\circ}\]