How to Solve Trigonometric Equations?

A step-by-step guide to solving trigonometric equations

Trigonometric equations are similar to algebraic equations and can be linear equations, quadratic equations, or polynomial equations.

To solve trigonometric equations, we use some general results and solutions of trigonometric equations. These results are as follows:

• For any real numbers \(x\) and \(y\), \(sin\: x = sin\: y\) implies \(x=n\pi +\left(-1\right)^ny\), where \(n ∈ Z\).
• For any real numbers \(x\) and \(y\), \(cos\: x = cos\: y\) implies \(x = 2nπ ± y\), where \(n ∈ Z\).
• If \(x\) and \(y\) are not odd multiples of \(\frac{π}{2}\), then \(tan\: x = tan\: y\) implies \(x = nπ + y\), where \(n ∈ Z\).

Solving trigonometric equations

We have two types of solutions to the trigonometric equations:

• Principal Solution:

The initial values of angles for trigonometric functions are called principal solutions.

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• General Solution: 

The values of the angles for the same answer of the trigonometric function are called the general solution of the trigonometric function. The general solutions of \(sin\:θ\), \(cos\:θ\), and \(tan\:θ\) are as follows.

• \(sin\:θ = sin\:α\), and the general solution is \(θ=n\pi +\left(-1\right)^nα\), where \(n ∈ Z\)
• \(cos\:θ = cos\:α\), and the general solution is \(θ = 2nπ + α\), where \(n ∈ Z\)
• \(tan\:θ = tan\:α\), and the general solution is \(θ = nπ + α\), where \(n ∈ Z\)

Steps to solve trigonometric equations

To solve a trigonometric equation, the following steps should be followed.

• Convert the given trigonometric equation to an equation with a single trigonometric ratio (sin, cos, tan).
• Change the equation with the trigonometric equation, having multiple angles, or submultiple angles into a simple angle.
• Now express the equation as a polynomial equation, a quadratic equation, or a linear equation.
• Solve the trigonometric equation similar to normal equations and find the value of the trigonometric ratio.
• The angle of the trigonometric ratio or the value of the trigonometric ratio shows the solution of the trigonometric equation.

Solving Trigonometric Equations – Example 1:

Find the principal solutions of the equation \(tan x=-\frac{1}{\sqrt{3}}\).

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Solution:

We know that \(tan\left(\frac{\pi }{6}\right)=\frac{1}{\sqrt{3}}\)

\(tan\:\left(\pi \:-\:\frac{\pi }{6}\:\right)=-tan\:\left(\frac{\pi }{6}\right)=-\:\frac{1}{\sqrt{3}}\)

\(tan\:\left(2\pi \:-\:\frac{\pi }{6}\right)=-tan\left(\frac{\pi }{6}\right)=-\:\frac{1}{\sqrt{3}}\)

So, the principal solutions are \(tan\: (π – \frac{π}{6}) = tan\: (\frac{5π}{6})\) and \(tan\: (2π – \frac{π}{6}) = tan\: (\frac{11π}{6})\).

Exercises for Solving Trigonometric Equations

Solve each trigonometric function for all possible values in degree.

1. \(\color{blue}{cos\:x+\sqrt{3}=-cos\:x}\)
2. \(\color{blue}{sin\:2x-\frac{\sqrt{3}}{2}=0}\)
3. \(\color{blue}{4\:sin\:\theta -1=2\:sin\:\theta +1}\)
4. \(\color{blue}{csc\:x+cot\:x=1}\)

1. \(\color{blue}{\:x=150^{\circ \:}+360^{\circ \:}n,\:x=210^{\circ \:}+360^{\circ \:}n}\)
2. \(\color{blue}{x=30^{\circ \:}+180^{\circ \:}n,\:x=60^{\circ \:}+180^{\circ \:}n}\)
3. \(\color{blue}{\:θ=90^{\circ \:}+360^{\circ \:}n}\)
4. \(\color{blue}{\:x=90^{\circ \:}+360^{\circ \:}n}\)