Trig equations with factoring worksheet

Treat sine or cosine as the variable and it is a quadratic like any other. When both appear squared, one Pythagorean substitution gets everything back into a single ratio first.

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Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name Trig Equations With Factoring Date Period Solve each equation on the interval [0, 2\pi). $2 \cos x \sin x + \cos x = 0$ $4 \cos^{2} x - 2 = 0$ $2 \cos x \sin x - \cos x = 0$ $2 \sin x \cos x - \sin x = 0$ $2 \sin x \cos x - \sqrt{3} \sin x = 0$ $4 \sin^{2} x - 2 = 0$ $2 \sin x \cos x + \sqrt{3} \sin x = 0$ $4 \sin^{2} x - 3 = 0$ $2 \cos x \sin x + \sqrt{3} \cos x = 0$ $4 \sin^{2} x - 4 = 0$ $4 \sin^{2} x - 1 = 0$ $4 \cos^{2} x - 1 = 0$ $4 \cos^{2} x - 4 = 0$ $2 \sin x \cos x + \sin x = 0$ $2 \cos x \sin x - \sqrt{3} \cos x = 0$ $4 \cos^{2} x - 3 = 0$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Trig Equations With Factoring

Date Period

Solve each equation on the interval [0, 2\pi).

$2 \cos x \sin x + \cos x = 0$
$4 \cos^{2} x - 2 = 0$
$2 \cos x \sin x - \cos x = 0$
$2 \sin x \cos x - \sin x = 0$
$2 \sin x \cos x - \sqrt{3} \sin x = 0$
$4 \sin^{2} x - 2 = 0$
$2 \sin x \cos x + \sqrt{3} \sin x = 0$
$4 \sin^{2} x - 3 = 0$
$2 \cos x \sin x + \sqrt{3} \cos x = 0$
$4 \sin^{2} x - 4 = 0$
$4 \sin^{2} x - 1 = 0$
$4 \cos^{2} x - 1 = 0$
$4 \cos^{2} x - 4 = 0$
$2 \sin x \cos x + \sin x = 0$
$2 \cos x \sin x - \sqrt{3} \cos x = 0$
$4 \cos^{2} x - 3 = 0$



Sigma Prep · sigmaprep.io/worksheets Name Trig Equations With FactoringAnswer keyVersion 1 Date Period Solve each equation on the interval [0, 2\pi). $2 \cos x \sin x + \cos x = 0$ $\frac{π}{2}, \frac{7 π}{6}, \frac{3 π}{2}, \frac{11 π}{6}$ $4 \cos^{2} x - 2 = 0$ $\frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4}$ $2 \cos x \sin x - \cos x = 0$ $\frac{π}{6}, \frac{π}{2}, \frac{5 π}{6}, \frac{3 π}{2}$ $2 \sin x \cos x - \sin x = 0$ $0, \frac{π}{3}, π, \frac{5 π}{3}$ $2 \sin x \cos x - \sqrt{3} \sin x = 0$ $0, \frac{π}{6}, π, \frac{11 π}{6}$ $4 \sin^{2} x - 2 = 0$ $\frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4}$ $2 \sin x \cos x + \sqrt{3} \sin x = 0$ $0, \frac{5 π}{6}, π, \frac{7 π}{6}$ $4 \sin^{2} x - 3 = 0$ $\frac{π}{3}, \frac{2 π}{3}, \frac{4 π}{3}, \frac{5 π}{3}$ $2 \cos x \sin x + \sqrt{3} \cos x = 0$ $\frac{π}{2}, \frac{4 π}{3}, \frac{3 π}{2}, \frac{5 π}{3}$ $4 \sin^{2} x - 4 = 0$ $\frac{π}{2}, \frac{3 π}{2}$ $4 \sin^{2} x - 1 = 0$ $\frac{π}{6}, \frac{5 π}{6}, \frac{7 π}{6}, \frac{11 π}{6}$ $4 \cos^{2} x - 1 = 0$ $\frac{π}{3}, \frac{2 π}{3}, \frac{4 π}{3}, \frac{5 π}{3}$ $4 \cos^{2} x - 4 = 0$ $0, π$ $2 \sin x \cos x + \sin x = 0$ $0, \frac{2 π}{3}, π, \frac{4 π}{3}$ $2 \cos x \sin x - \sqrt{3} \cos x = 0$ $\frac{π}{3}, \frac{π}{2}, \frac{2 π}{3}, \frac{3 π}{2}$ $4 \cos^{2} x - 3 = 0$ $\frac{π}{6}, \frac{5 π}{6}, \frac{7 π}{6}, \frac{11 π}{6}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Trig Equations With FactoringAnswer keyVersion 1

Date Period

Solve each equation on the interval [0, 2\pi).

$2 \cos x \sin x + \cos x = 0$ $\frac{π}{2}, \frac{7 π}{6}, \frac{3 π}{2}, \frac{11 π}{6}$
$4 \cos^{2} x - 2 = 0$ $\frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4}$
$2 \cos x \sin x - \cos x = 0$ $\frac{π}{6}, \frac{π}{2}, \frac{5 π}{6}, \frac{3 π}{2}$
$2 \sin x \cos x - \sin x = 0$ $0, \frac{π}{3}, π, \frac{5 π}{3}$
$2 \sin x \cos x - \sqrt{3} \sin x = 0$ $0, \frac{π}{6}, π, \frac{11 π}{6}$
$4 \sin^{2} x - 2 = 0$ $\frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4}$
$2 \sin x \cos x + \sqrt{3} \sin x = 0$ $0, \frac{5 π}{6}, π, \frac{7 π}{6}$
$4 \sin^{2} x - 3 = 0$ $\frac{π}{3}, \frac{2 π}{3}, \frac{4 π}{3}, \frac{5 π}{3}$
$2 \cos x \sin x + \sqrt{3} \cos x = 0$ $\frac{π}{2}, \frac{4 π}{3}, \frac{3 π}{2}, \frac{5 π}{3}$
$4 \sin^{2} x - 4 = 0$ $\frac{π}{2}, \frac{3 π}{2}$
$4 \sin^{2} x - 1 = 0$ $\frac{π}{6}, \frac{5 π}{6}, \frac{7 π}{6}, \frac{11 π}{6}$
$4 \cos^{2} x - 1 = 0$ $\frac{π}{3}, \frac{2 π}{3}, \frac{4 π}{3}, \frac{5 π}{3}$
$4 \cos^{2} x - 4 = 0$ $0, π$
$2 \sin x \cos x + \sin x = 0$ $0, \frac{2 π}{3}, π, \frac{4 π}{3}$
$2 \cos x \sin x - \sqrt{3} \cos x = 0$ $\frac{π}{3}, \frac{π}{2}, \frac{2 π}{3}, \frac{3 π}{2}$
$4 \cos^{2} x - 3 = 0$ $\frac{π}{6}, \frac{5 π}{6}, \frac{7 π}{6}, \frac{11 π}{6}$

How to do these

$2 \sin^{2} x - \sin x - 1 = 0$

It factors as (2 sin x + 1)(sin x - 1).
So sine is minus one half, or 1.
That is three angles on one turn.

The answer is $\frac{π}{2}, \frac{7 π}{6}, \frac{11 π}{6}$ .

Where students go wrong

Dividing both sides by sin x to get rid of it. That throws away every solution where sine is zero. Factor it out instead and set both factors to zero.

Also called factoring trig equations, quadratic trig equations or Pythagorean identity.

Questions about these worksheets

Yes. Every trig equations with factoring sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the trig equations with factoring questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

Dividing both sides by sin x to get rid of it. That throws away every solution where sine is zero. Factor it out instead and set both factors to zero.

Trig equations with factoring worksheet

Select your difficulty

Trig Equations With Factoring

Trig Equations With FactoringAnswer keyVersion 1

How to do these

Where students go wrong

Questions about these worksheets

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