Solving trigonometric equations

​​In a nutshell

You have already learnt how to solve trigonometric equations in degrees. Trigonometric equations can be solved in radians using a similar method.

Change angle units on a calculator

You can solve trigonometric equations as you have already done, however make sure to set a calculator in 'radians' mode as follows:

CALCULATOR TIP

Select 'Shift' and 'SETUP'. The display should show:

​$$\boxed{\begin{aligned}1&: Input/Output \\2&: Angle \ Unit \\3&: Number \ format \\4&: Engineer \ Symbol\end{aligned}}$$​​

Select option $$2$$ to change the angle units. The display will show:

$$\boxed{\begin{aligned}1&: Degree \\2&: Radian \\3&: Gradian\end{aligned}}$$​​

Select either $$1$$ for degrees or $$2$$ for radians.​

Solve trigonometric equations

Trigonometric equations in radians can now be solved in the same way as solving equations in degrees. Once you have calculated the 'principle value' after using the inverse trig function on a calculator, find the other values in the range either by using the appropriate trigonometric graph with the angles on the $$x$$-axis given in radians, or by using the following to work out the other values:

Note: There are usually two answers between $$0$$ and $$2 \pi$$ radians. The first answer is the principal value from a calculator, the second value can be found by using the symmetry of a trigonometric graph (used in the table above). Any other values can be found by taking the two answers and either adding or subtracting $$2 \pi$$ from the first two answers.

Example 1

Solve $$2 \sin 3\theta = 1$$. Give all answers in the range $$0 \le \theta \le 2 \pi$$.

Rearrange for $$\sin 3 \theta$$ and solve:

$$\begin{aligned}2 \sin (3 \theta) &= 1 \\\\\sin (3 \theta) &= \dfrac 1 2 \\\\ 3 \theta &= \sin^{-1} \Big( \dfrac 1 2 \Big ) \\\\3 \theta &= \dfrac {\pi}{6}\end {aligned}$$​​

This gives the principal value of $$\dfrac {\pi}{6}$$. Adjust the range given and work out the other values. 

The range for $$\theta$$ is given as:

$$0 \le \theta \le 2 \pi$$​​

Adjust the range for $$3 \theta$$:

​$$0 \le 3\theta \le 6 \pi$$​​

Working out all values of $$3 \theta$$ up to $$6 \pi$$ using a graph or alternative method:

$$3 \theta = \dfrac {\pi}{6 }, \underbrace{\dfrac {5 \pi}{6 }}_{\pi - \frac {\pi}{6}}, \underbrace{\dfrac {13 \pi}{6 }}_{\frac {\pi}{6 }+2 \pi}, \underbrace{\dfrac {17 \pi}{6 }}_{\frac {5 \pi}{6 }+2 \pi}, \underbrace{\dfrac {25 \pi}{6 }}_{\frac {13 \pi}{6 }+2 \pi}, \underbrace {\dfrac {29 \pi}{6 }}_{\frac {17 \pi}{6 }+2 \pi}$$​​

Divide by $$3$$ to find all values of $$\theta$$: 

​$$\underline{\theta = \dfrac {\pi}{18 }, \dfrac {5 \pi}{18 }, \dfrac {13 \pi}{18 }, \dfrac {17 \pi}{18 }, \dfrac {25 \pi}{18 }, \dfrac {29 \pi}{18 }}$$​​