How to Solve Trigonometric Equations?

Q: What is the general solution of a trigonometric equation?

The general solution lists every angle that works by adding whole multiples of the period. Sine and cosine repeat every 2π, while tangent repeats every π. For tan(x) = √3, the solution in one cycle is x = π/3, so the general solution is x = π/3 + πk, where k is any integer.

Q: What is the principal solution of a trigonometric equation?

Principal solutions are the solutions that fall in one full turn, usually 0 ≤ x < 2π. For sin(x) = −1/2, the principal solutions are x = 7π/6 and x = 11π/6, since sine is negative in quadrants III and IV with a reference angle of π/6. Your course may use a different interval, so check which one your teacher or textbook means.

Q: How do you solve trigonometric equations without a calculator?

Get the equation into a form like sin(x) = 1/2 or cos(x) = −√2/2, then use the special angle values from the unit circle. Find the reference angle, then decide which quadrants match the sign. For cos(x) = −√2/2, the reference angle is π/4, and cosine is negative in quadrants II and III, so x = 3π/4 or 5π/4.

Q: How do you solve trigonometric equations with multiple angles?

Solve for the whole angle first, then divide. For cos(2x) = 0, set 2x = π/2 + πk, so x = π/4 + πk/2. Because the angle is doubled, more solutions fit in one cycle: in 0 ≤ x < 2π you get π/4, 3π/4, 5π/4 and 7π/4. Always divide the period term too, not just the first angle.

TL;DR: Solving a trig equation is a three-step rhythm. First, isolate the trig function on one side just like you'd isolate x in regular algebra. Next, find every reference-angle solution that lives in a single revolution (0 to 2 pi). Finally, tack on "plus 2 pi times any integer k" to capture every other rotation. One heads-up: if you squared both sides anywhere along the way, plug your answers back in, squaring can sneak in extra solutions that don't actually work.

Key takeaways:

Isolate the trig function first, just like solving algebra equations.
Use the unit circle (or inverse trig functions) to find principal solutions.
Sin and csc repeat every \(2\pi\); tan and cot repeat every \(\pi\). Adjust your general solution accordingly.
When the equation involves powers or compound expressions, use identities to reduce to a single trig function.
Always verify final answers in the original equation, squaring can introduce extraneous solutions.

A trigonometric equation is an equation whose variable is expressed in terms of the value of the trigonometric function. In this step-by-step guide, you will learn more about trigonometric equations and solving them.

Tutor-style math help

Solve Trigonometric Equations: what to notice and how to work it

Trigonometry skill

Trigonometry connects an angle to a triangle ratio, a unit-circle coordinate, or a repeating graph. Choosing the right picture makes the problem much easier.

What to notice first

Decide whether the problem is triangle-based, circle-based, or graph-based. Then use the matching definition.

Common student mistake

Do not mix degrees and radians. The angle unit must match the formula, graph scale, or calculator setting.

Key formulas and cues

\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\)

\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)

\(\tan\theta=\frac{\sin\theta}{\cos\theta}\)

\(\sin^2\theta+\cos^2\theta=1\)

A reliable path

Choose the modelUse a right triangle, the unit circle, or a transformed graph.
Track unitsConvert degrees and radians when needed.
Use identitiesReplace complicated trig expressions with equivalent simpler ones.

Worked examples

Right-triangle sine

Example: opposite = 5, hypotenuse = 13

Sine is opposite over hypotenuse.
Substitute 5 and 13.
Leave the ratio simplified.

Answer: \(\sin\theta=\frac5{13}\)

Unit-circle cosine

Example: \(\cos(0)\)

At angle 0, the point is (1, 0).
Cosine is the x-coordinate.
Read the x-value.

Answer: \(1\)

Open pop-up practice

Try one before moving on

Try: In a right triangle, tangent equals which ratio?

Answer: Opposite over adjacent.

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

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.

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.

Trigonometry for Beginners The Ultimate Step by Step Guide to Acing Trigonometry

Solving Trigonometric Equations, Example 1:

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

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.

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

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

Recommended EffortlessMath Books

For a workbook that builds trigonometric equations into a full trig unit, the Trigonometry for Beginners (resource currently unavailable) walks through every solution technique with worked examples. For pre-calc-level coverage that prepares you for calculus, see the Pre-Calculus for Beginners.

Frequently Asked Questions

What is a trigonometric equation?

A trigonometric equation is an equation where the unknown is an angle inside a trig function, such as 2sin(x) − 1 = 0 or tan(x) = √3. Unlike an identity, it is true only for certain angles. Because trig functions repeat, most trig equations have infinitely many solutions, which is why answers are often written with + 2πk or + πk.

How do you solve trigonometric equations step by step?

First isolate the trig function with algebra. Next, use the unit circle or an inverse trig function to find every solution in one cycle, 0 ≤ x < 2π. Then add the period times an integer k. For 2cos(x) = √3, cos(x) = √3/2, so x = π/6 or 11π/6, and the general solution is x = π/6 + 2πk or x = 11π/6 + 2πk.

What is the general solution of a trigonometric equation?

The general solution lists every angle that works by adding whole multiples of the period. Sine and cosine repeat every 2π, while tangent repeats every π. For tan(x) = √3, the solution in one cycle is x = π/3, so the general solution is x = π/3 + πk, where k is any integer.

What is the principal solution of a trigonometric equation?

Principal solutions are the solutions that fall in one full turn, usually 0 ≤ x < 2π. For sin(x) = −1/2, the principal solutions are x = 7π/6 and x = 11π/6, since sine is negative in quadrants III and IV with a reference angle of π/6. Your course may use a different interval, so check which one your teacher or textbook means.

How do you solve trigonometric equations without a calculator?

Get the equation into a form like sin(x) = 1/2 or cos(x) = −√2/2, then use the special angle values from the unit circle. Find the reference angle, then decide which quadrants match the sign. For cos(x) = −√2/2, the reference angle is π/4, and cosine is negative in quadrants II and III, so x = 3π/4 or 5π/4.

How do you solve trigonometric equations with multiple angles?

Solve for the whole angle first, then divide. For cos(2x) = 0, set 2x = π/2 + πk, so x = π/4 + πk/2. Because the angle is doubled, more solutions fit in one cycle: in 0 ≤ x < 2π you get π/4, 3π/4, 5π/4 and 7π/4. Always divide the period term too, not just the first angle.

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