Simple trigonometric equations

​​In a nutshell

You need to learn how to solve equations of the form $$\sin(\theta) = k$$ and $$\cos(\theta) = k$$​ for $$-1 \leq k \leq1$$, or $$\tan(\theta) = p$$​ for $$p \in \mathbb{R}$$​, over a given interval of $$\theta$$​.

Ranges of trigonometric functions

For any value of $$\theta$$​, $$-1 \leq \sin(\theta) \leq 1$$​ and $$-1 \leq \cos(\theta) \leq 1$$​, so solutions will only ever exist for $$\sin(\theta) = k$$​ or $$\cos(\theta) = k$$​ when $$-1 \leq k \leq 1$$.

$$\tan(\theta)$$​ takes on all values in $$\mathbb{R}$$​ as $$\theta$$​ varies from $$0^{\circ}$$​ to $$180^{\circ}$$​, so solutions to $$\tan(\theta) = p$$​ exist over some interval for any value of $$p$$.

Multiple solutions may exist in a given interval.

The inverse trigonometric functions on your calculator can be used to find a single solution to equations of this type. The output they give is known as the "principal value". The principal values lie in the following ranges:

Finding solutions

Given a single solution to $$\cos(\theta) = k$$​, $$\sin(\theta) = k$$​, or $$\tan(\theta) = p$$​ for $$-1 \le k \le 1$$​ or $$p \in \mathbb{R}$$ in the interval $$0\degree \le \theta \le 360\degree$$, you can find all other solutions in the same interval via the following table, which gives other values of $$\theta$$​ solving the relevant equation in the same interval.

This can be used to find solutions in any interval by adding or subtracting multiples of $$360\degree$$​ to your solutions.

Solving simple trigonometric equations

The following procedure will help you to identify all solutions to equations of the form $$\sin(\theta) = k$$​ and $$\cos(\theta) = k$$​ for $$-1 \leq k \leq 1$$, or $$\tan(\theta) = p$$​ for any real value of $$p$$:

PROCEDURE

Example 1

What are all the solutions to the equation $$\cos(\theta) = \dfrac12$$ in the interval $$-180^{\circ} \leq \theta \leq 180^{\circ}?$$

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First, find the principal value by taking the inverse cosine:

$$\theta=\cos^{-1}(\dfrac{1}{2})=60\degree$$​​

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Check if there are any other solutions in the interval $$0\degree \le \theta \le 360\degree$$ which are less than or equal to $$180\degree$$:

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​$$360\degree - 60\degree = 300\degree > 180\degree$$, so there are not.

Now, check if there are any other solutions in the interval $$-360\degree \le \theta \le 0\degree$$ by subtracting $$360\degree$$ from both $$\theta$$ and $$360\degree - \theta$$ to get:

$$\begin{aligned}\theta - 360\degree &= 60\degree - 360\degree\\360\degree - \theta &= -300\degree\end{aligned}$$​​

 $$\begin{aligned}360\degree -\theta - 360\degree &= -\theta\\-\theta &=-60\degree\end{aligned}$$​

The only one of these which lies in the interval $$-180\degree \le \theta \le 180\degree$$ is $$-60\degree$$.

Therefore, the solutions to the equation $$\cos(\theta) = 60\degree$$ in the interval $$-180\degree \le \theta \le 180\degree$$ are $$\underline{\theta = 60\degree}$$ and $$\underline{\theta = -60\degree}$$​

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Example 2

What are all the solutions to the equation $$-4\sin(\theta) = 3$$ in the interval $$0 \leq \theta \leq 360^{\circ}$$? Give your answers to $$3\space s.f.$$.

Rearrange the equation to see that:

$$\begin{aligned}-4\sin(\theta) &=3\\\sin(\theta) &= -\dfrac34\end{aligned}$$​​

Find the principal value by taking the inverse sine:

​$$\theta=\sin^{-1}(-\dfrac{3}{4})=-48.590377...$$​​

Sketch the relevant graph:

This is the graph of $$y = \sin(x)$$ and the line $$y = -\dfrac34$$. See from its symmetries that the only solutions which lie in the desired interval are $$180^\circ - \sin^{-1}\left(-\dfrac34\right)^\circ$$, and $$360^{\circ} + \sin^{-1}\left(-\dfrac34\right)^\circ$$. To $$3$$ significant figures, these work out to $$229^{\circ}$$, and $$311^\circ$$.

Therefore, the solutions to the equation $$-4\sin(\theta) = 3$$ which lie in the interval $$0^\circ \leq \theta \leq 360^\circ$$ are $$\underline{\theta = 229^\circ}$$ and $$\underline{\theta = 311^\circ}(3 \ s.f.)$$ 

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