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Simple trigonometric equations

Simple trigonometric equations

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Explainer Video

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Summary

Simple trigonometric equations

​​In a nutshell

You need to learn how to solve equations of the form sin⁡(θ)=k\sin(\theta) = ksin(θ)=k and cos⁡(θ)=k\cos(\theta) = kcos(θ)=k​ for −1≤k≤1-1 \leq k \leq1 −1≤k≤1, or tan⁡(θ)=p\tan(\theta) = ptan(θ)=p​ for p∈Rp \in \mathbb{R}p∈R​, over a given interval of θ\thetaθ​.



Ranges of trigonometric functions

For any value of θ\thetaθ​, −1≤sin⁡(θ)≤1-1 \leq \sin(\theta) \leq 1−1≤sin(θ)≤1​ and −1≤cos⁡(θ)≤1-1 \leq \cos(\theta) \leq 1−1≤cos(θ)≤1​, so solutions will only ever exist for sin⁡(θ)=k\sin(\theta) = ksin(θ)=k​ or cos⁡(θ)=k\cos(\theta) = kcos(θ)=k​ when −1≤k≤1-1 \leq k \leq 1−1≤k≤1.

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

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:


FUNCTION

OUTPUT RANGE

cos⁡−1\cos^{-1}cos−1​​

​0∘≤θ≤180∘0^\circ \leq \theta \leq 180^{\circ}0∘≤θ≤180∘​​

sin⁡−1\sin^{-1}sin−1​​

​−90∘≤θ≤90∘-90^\circ \leq \theta \leq 90^\circ−90∘≤θ≤90∘​​

​tan⁡−1\tan^{-1}tan−1​​

​−90∘≤θ≤90∘-90^\circ \leq \theta \leq 90^\circ−90∘≤θ≤90∘​​



Finding solutions

Given a single solution to cos⁡(θ)=k\cos(\theta) = kcos(θ)=k​, sin⁡(θ)=k\sin(\theta) = ksin(θ)=k​, or tan⁡(θ)=p\tan(\theta) = ptan(θ)=p​ for −1≤k≤1-1 \le k \le 1−1≤k≤1​ or p∈Rp \in \mathbb{R}p∈R in the interval 0°≤θ≤360°0\degree \le \theta \le 360\degree0°≤θ≤360°, you can find all other solutions in the same interval via the following table, which gives other values of θ\thetaθ​ solving the relevant equation in the same interval.

equation
​0°≤θ≤360°0\degree \le \theta \le 360\degree0°≤θ≤360°​​
​cos⁡(θ)=k,−1≤k≤1\cos(\theta) = k, -1 \le k \le 1cos(θ)=k,−1≤k≤1​​
360°−θ360\degree - \theta360°−θ​​
​sin⁡(θ)=k,−1≤k≤1\sin(\theta) = k, -1 \le k \le 1sin(θ)=k,−1≤k≤1​​
​180°−θ180\degree - \theta180°−θ​​
​tan⁡(θ)=p,p∈R\tan(\theta) = p, p \in \mathbb{R}tan(θ)=p,p∈R​​
​180°+θ180\degree + \theta180°+θ​​

This can be used to find solutions in any interval by adding or subtracting multiples of 360°360\degree360°​ to your solutions.



Solving simple trigonometric equations

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

PROCEDURE

1.

Use the appropriate inverse trigonometric function to obtain the principal value.

2.

(OPTIONAL) Sketch the graph of the appropriate trigonometric function so that the domain covers the interval you are looking for solutions in, and contains the principal value.

3.

Either use the above table, or use the symmetries of the graph of the function you sketched, to read off all possible solutions in the desired interval from your sketch.



Example 1

What are all the solutions to the equation cos⁡(θ)=12\cos(\theta) = \dfrac12cos(θ)=21​ in the interval −180∘≤θ≤180∘?-180^{\circ} \leq \theta \leq 180^{\circ}?−180∘≤θ≤180∘?

​

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

θ=cos⁡−1(12)=60°\theta=\cos^{-1}(\dfrac{1}{2})=60\degreeθ=cos−1(21​)=60°​​


​​

Check if there are any other solutions in the interval 0°≤θ≤360°0\degree \le \theta \le 360\degree0°≤θ≤360° which are less than or equal to 180°180\degree180°:

​

​360°−60°=300°>180°360\degree - 60\degree = 300\degree > 180\degree360°−60°=300°>180°, so there are not.


Now, check if there are any other solutions in the interval −360°≤θ≤0°-360\degree \le \theta \le 0\degree−360°≤θ≤0° by subtracting 360°360\degree360° from both θ\thetaθ and 360°−θ360\degree - \theta360°−θ to get:


θ−360°=60°−360°360°−θ=−300°\begin{aligned}\theta - 360\degree &= 60\degree - 360\degree\\360\degree - \theta &= -300\degree\end{aligned}θ−360°360°−θ​=60°−360°=−300°​​​


 360°−θ−360°=−θ−θ=−60°\begin{aligned}360\degree -\theta - 360\degree &= -\theta\\-\theta &=-60\degree\end{aligned}360°−θ−360°−θ​=−θ=−60°​​

The only one of these which lies in the interval −180°≤θ≤180°-180\degree \le \theta \le 180\degree−180°≤θ≤180° is −60°-60\degree−60°.


Therefore, the solutions to the equation cos⁡(θ)=60°\cos(\theta) = 60\degreecos(θ)=60° in the interval −180°≤θ≤180°-180\degree \le \theta \le 180\degree−180°≤θ≤180° are θ=60°‾\underline{\theta = 60\degree}θ=60°​ and θ=−60°‾\underline{\theta = -60\degree}θ=−60°​​


​

Example 2

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


Rearrange the equation to see that:

−4sin⁡(θ)=3sin⁡(θ)=−34\begin{aligned}-4\sin(\theta) &=3\\\sin(\theta) &= -\dfrac34\end{aligned}−4sin(θ)sin(θ)​=3=−43​​​​


Find the principal value by taking the inverse sine:

​θ=sin⁡−1(−34)=−48.590377...\theta=\sin^{-1}(-\dfrac{3}{4})=-48.590377...θ=sin−1(−43​)=−48.590377...​​


Sketch the relevant graph:

Maths; Trigonometric equations; KS5 Year 12; Simple trigonometric equations


This is the graph of y=sin⁡(x)y = \sin(x)y=sin(x) and the line y=−34y = -\dfrac34y=−43​. See from its symmetries that the only solutions which lie in the desired interval are 180∘−sin⁡−1(−34)∘180^\circ - \sin^{-1}\left(-\dfrac34\right)^\circ180∘−sin−1(−43​)∘, and 360∘+sin⁡−1(−34)∘360^{\circ} + \sin^{-1}\left(-\dfrac34\right)^\circ360∘+sin−1(−43​)∘. To 333 significant figures, these work out to 229∘229^{\circ}229∘, and 311∘311^\circ311∘.


Therefore, the solutions to the equation −4sin⁡(θ)=3-4\sin(\theta) = 3−4sin(θ)=3 which lie in the interval 0∘≤θ≤360∘0^\circ \leq \theta \leq 360^\circ0∘≤θ≤360∘ are θ=229∘‾\underline{\theta = 229^\circ}θ=229∘​ and θ=311∘‾(3 s.f.)\underline{\theta = 311^\circ}(3 \ s.f.)θ=311∘​(3 s.f.) 



​​​

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FAQs - Frequently Asked Questions

How many solutions are there to a simple trigonometric function?

Multiple solutions may exist in a given interval.

When can I solve a simple trigonometric equation?

Solutions will only ever exist for sin(x) = k or cos(x) = k when k is between -1 and 1. tan(x) takes on all real values as x varies, so solutions to tan(x) = p exist for any value of p.

How do I solve simple trigonometric equations?

The inverse trigonometric functions on your calculator can be used to find a single solution to equations of this type.

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