Double angle identities worksheet for Precalculus

In Precalculus about half of these give one ratio and a quadrant, so the first job is finding the other ratio with the right sign. The other half give both sine and cosine and ask for tan 2θ. Two questions use the half-angle formula for exact values, such as cos 105° as half of 210°.

Also taught in Algebra 2, at its own level and with its own examples.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Double Angle Identities Date Period Evaluate each. Use the double-angle identities Given that $\cos θ = - \frac{60}{61}$ , with $θ$ in quadrant II, find $\tan 2 θ$ . $\cos θ = - \frac{3}{5}$ and $θ$ is in Quadrant III. Find $\sin 2 θ$ If $\sin θ = - \frac{24}{25}$ and $θ$ is in quadrant IV, find $\cos 2 θ$ . $\sin θ = - \frac{21}{29}$ and $θ$ is in Quadrant III. Find $\sin 2 θ$ For an angle $θ$ in quadrant III with $\tan θ = \frac{12}{35}$ , find $\sin 2 θ$ . $\sin θ = - \frac{15}{17}$ and $θ$ is in Quadrant IV. Find $\cos 2 θ$ $\cos θ = \frac{21}{29}$ and $θ$ is in Quadrant IV. Find $\cos 2 θ$ If $\sin θ = - \frac{48}{73}$ and $θ$ is in quadrant IV, find $\sin 2 θ$ . $\sin θ = - \frac{5}{13}$ and $θ$ is in Quadrant III. Find $\tan 2 θ$ Given that $\cos θ = - \frac{28}{53}$ , with $θ$ in quadrant II, find $\cos 2 θ$ . $\cos θ = - \frac{21}{29}$ and $θ$ is in Quadrant III. Find $\tan 2 θ$ For an angle $θ$ in quadrant III with $\tan θ = \frac{56}{33}$ , find $\tan 2 θ$ . Given that $\sin θ = \frac{63}{65}$ , with $θ$ in quadrant II, find $\tan 2 θ$ . $\sin θ = \frac{3}{5}$ and $θ$ is in Quadrant II. Find $\tan 2 θ$ If $\tan θ = - \frac{11}{60}$ and $θ$ is in quadrant II, find $\cos 2 θ$ . $\cos θ = \frac{15}{17}$ and $θ$ is in Quadrant IV. Find $\sin 2 θ$ For an angle $θ$ in quadrant III with $\sin θ = - \frac{40}{41}$ , find $\cos 2 θ$ . $\sin θ = - \frac{7}{25}$ and $θ$ is in Quadrant IV. Find $\tan 2 θ$ Find exact values with half angles Find $\sin 165^{\circ}$ exactly Find $\cos {67.5}^{\circ}$ exactly Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Double Angle Identities

Date Period

Evaluate each.

Use the double-angle identities
Given that $\cos θ = - \frac{60}{61}$ , with $θ$ in quadrant II, find $\tan 2 θ$ .
$\cos θ = - \frac{3}{5}$ and $θ$ is in Quadrant III. Find $\sin 2 θ$
If $\sin θ = - \frac{24}{25}$ and $θ$ is in quadrant IV, find $\cos 2 θ$ .
$\sin θ = - \frac{21}{29}$ and $θ$ is in Quadrant III. Find $\sin 2 θ$
For an angle $θ$ in quadrant III with $\tan θ = \frac{12}{35}$ , find $\sin 2 θ$ .
$\sin θ = - \frac{15}{17}$ and $θ$ is in Quadrant IV. Find $\cos 2 θ$
$\cos θ = \frac{21}{29}$ and $θ$ is in Quadrant IV. Find $\cos 2 θ$
If $\sin θ = - \frac{48}{73}$ and $θ$ is in quadrant IV, find $\sin 2 θ$ .
$\sin θ = - \frac{5}{13}$ and $θ$ is in Quadrant III. Find $\tan 2 θ$
Given that $\cos θ = - \frac{28}{53}$ , with $θ$ in quadrant II, find $\cos 2 θ$ .
$\cos θ = - \frac{21}{29}$ and $θ$ is in Quadrant III. Find $\tan 2 θ$
For an angle $θ$ in quadrant III with $\tan θ = \frac{56}{33}$ , find $\tan 2 θ$ .
Given that $\sin θ = \frac{63}{65}$ , with $θ$ in quadrant II, find $\tan 2 θ$ .
$\sin θ = \frac{3}{5}$ and $θ$ is in Quadrant II. Find $\tan 2 θ$
If $\tan θ = - \frac{11}{60}$ and $θ$ is in quadrant II, find $\cos 2 θ$ .
$\cos θ = \frac{15}{17}$ and $θ$ is in Quadrant IV. Find $\sin 2 θ$
For an angle $θ$ in quadrant III with $\sin θ = - \frac{40}{41}$ , find $\cos 2 θ$ .
$\sin θ = - \frac{7}{25}$ and $θ$ is in Quadrant IV. Find $\tan 2 θ$

Find exact values with half angles
Find $\sin 165^{\circ}$ exactly
Find $\cos {67.5}^{\circ}$ exactly



Sigma Prep · sigmaprep.io/worksheets Name Double Angle IdentitiesAnswer keyVersion 2 Date Period Evaluate each. Use the double-angle identities Given that $\cos θ = - \frac{60}{61}$ , with $θ$ in quadrant II, find $\tan 2 θ$ . $- \frac{1320}{3479}$ $\cos θ = - \frac{3}{5}$ and $θ$ is in Quadrant III. Find $\sin 2 θ$ $\frac{24}{25}$ If $\sin θ = - \frac{24}{25}$ and $θ$ is in quadrant IV, find $\cos 2 θ$ . $- \frac{527}{625}$ $\sin θ = - \frac{21}{29}$ and $θ$ is in Quadrant III. Find $\sin 2 θ$ $\frac{840}{841}$ For an angle $θ$ in quadrant III with $\tan θ = \frac{12}{35}$ , find $\sin 2 θ$ . $\frac{840}{1369}$ $\sin θ = - \frac{15}{17}$ and $θ$ is in Quadrant IV. Find $\cos 2 θ$ $- \frac{161}{289}$ $\cos θ = \frac{21}{29}$ and $θ$ is in Quadrant IV. Find $\cos 2 θ$ $\frac{41}{841}$ If $\sin θ = - \frac{48}{73}$ and $θ$ is in quadrant IV, find $\sin 2 θ$ . $- \frac{5280}{5329}$ $\sin θ = - \frac{5}{13}$ and $θ$ is in Quadrant III. Find $\tan 2 θ$ $\frac{120}{119}$ Given that $\cos θ = - \frac{28}{53}$ , with $θ$ in quadrant II, find $\cos 2 θ$ . $- \frac{1241}{2809}$ $\cos θ = - \frac{21}{29}$ and $θ$ is in Quadrant III. Find $\tan 2 θ$ $\frac{840}{41}$ For an angle $θ$ in quadrant III with $\tan θ = \frac{56}{33}$ , find $\tan 2 θ$ . $- \frac{3696}{2047}$ Given that $\sin θ = \frac{63}{65}$ , with $θ$ in quadrant II, find $\tan 2 θ$ . $\frac{2016}{3713}$ $\sin θ = \frac{3}{5}$ and $θ$ is in Quadrant II. Find $\tan 2 θ$ $- \frac{24}{7}$ If $\tan θ = - \frac{11}{60}$ and $θ$ is in quadrant II, find $\cos 2 θ$ . $\frac{3479}{3721}$ $\cos θ = \frac{15}{17}$ and $θ$ is in Quadrant IV. Find $\sin 2 θ$ $- \frac{240}{289}$ For an angle $θ$ in quadrant III with $\sin θ = - \frac{40}{41}$ , find $\cos 2 θ$ . $- \frac{1519}{1681}$ $\sin θ = - \frac{7}{25}$ and $θ$ is in Quadrant IV. Find $\tan 2 θ$ $- \frac{336}{527}$ Find exact values with half angles Find $\sin 165^{\circ}$ exactly $\frac{\sqrt{6} - \sqrt{2}}{4}$ Find $\cos {67.5}^{\circ}$ exactly $\frac{\sqrt{2 - \sqrt{2}}}{2}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Double Angle IdentitiesAnswer keyVersion 2

Date Period

Evaluate each.

Use the double-angle identities
Given that $\cos θ = - \frac{60}{61}$ , with $θ$ in quadrant II, find $\tan 2 θ$ . $- \frac{1320}{3479}$
$\cos θ = - \frac{3}{5}$ and $θ$ is in Quadrant III. Find $\sin 2 θ$ $\frac{24}{25}$
If $\sin θ = - \frac{24}{25}$ and $θ$ is in quadrant IV, find $\cos 2 θ$ . $- \frac{527}{625}$
$\sin θ = - \frac{21}{29}$ and $θ$ is in Quadrant III. Find $\sin 2 θ$ $\frac{840}{841}$
For an angle $θ$ in quadrant III with $\tan θ = \frac{12}{35}$ , find $\sin 2 θ$ . $\frac{840}{1369}$
$\sin θ = - \frac{15}{17}$ and $θ$ is in Quadrant IV. Find $\cos 2 θ$ $- \frac{161}{289}$
$\cos θ = \frac{21}{29}$ and $θ$ is in Quadrant IV. Find $\cos 2 θ$ $\frac{41}{841}$
If $\sin θ = - \frac{48}{73}$ and $θ$ is in quadrant IV, find $\sin 2 θ$ . $- \frac{5280}{5329}$
$\sin θ = - \frac{5}{13}$ and $θ$ is in Quadrant III. Find $\tan 2 θ$ $\frac{120}{119}$
Given that $\cos θ = - \frac{28}{53}$ , with $θ$ in quadrant II, find $\cos 2 θ$ . $- \frac{1241}{2809}$
$\cos θ = - \frac{21}{29}$ and $θ$ is in Quadrant III. Find $\tan 2 θ$ $\frac{840}{41}$
For an angle $θ$ in quadrant III with $\tan θ = \frac{56}{33}$ , find $\tan 2 θ$ . $- \frac{3696}{2047}$
Given that $\sin θ = \frac{63}{65}$ , with $θ$ in quadrant II, find $\tan 2 θ$ . $\frac{2016}{3713}$
$\sin θ = \frac{3}{5}$ and $θ$ is in Quadrant II. Find $\tan 2 θ$ $- \frac{24}{7}$
If $\tan θ = - \frac{11}{60}$ and $θ$ is in quadrant II, find $\cos 2 θ$ . $\frac{3479}{3721}$
$\cos θ = \frac{15}{17}$ and $θ$ is in Quadrant IV. Find $\sin 2 θ$ $- \frac{240}{289}$
For an angle $θ$ in quadrant III with $\sin θ = - \frac{40}{41}$ , find $\cos 2 θ$ . $- \frac{1519}{1681}$
$\sin θ = - \frac{7}{25}$ and $θ$ is in Quadrant IV. Find $\tan 2 θ$ $- \frac{336}{527}$

Find exact values with half angles
Find $\sin 165^{\circ}$ exactly $\frac{\sqrt{6} - \sqrt{2}}{4}$
Find $\cos {67.5}^{\circ}$ exactly $\frac{\sqrt{2 - \sqrt{2}}}{2}$

How to do these

$\sin θ = - \frac{5}{13}$ and $θ$ is in Quadrant IV. Find $\sin 2 θ$

In Quadrant IV cosine is positive, so cos θ = 12/13.
sin 2θ = 2 sin θ cos θ = 2(-5/13)(12/13).

The answer is $- \frac{120}{169}$ .

Where students go wrong

Taking the sign of the missing ratio from the triangle instead of the quadrant. A 5-12-13 triangle gives 12/13, but whether cos θ is positive or negative depends on where θ sits.

Also called double angle formulas, sin 2 theta or cos 2 theta.

What you can put on this worksheet

Double angle from a given ratio: Given that $\cos θ = \frac{12}{37}$ and $\sin θ = \frac{35}{37}$ , find $\cos 2 θ$ . → $- \frac{1081}{1369}$
From a given ratio: If $\cos θ = \frac{8}{17}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ → $\frac{5 \sqrt{34}}{34}$
From one ratio and the quadrant: $\cos θ = - \frac{15}{17}$ and $θ$ is in Quadrant II. Find $\sin 2 θ$ → $- \frac{240}{289}$
Exact values of half a special angle: Find $\sin 75^{\circ}$ exactly → $\frac{\sqrt{6} + \sqrt{2}}{4}$
Reciprocal functions, given tan or cot, roots in the ratio: If $\sin θ = \frac{15}{17}$ and $θ$ is in quadrant II. Find $\cot \frac{θ}{2}$ → $\cot \frac{θ}{2} = \frac{3}{5}$

Questions about these worksheets

Yes. Every double angle identities 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 double angle identities 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.

You pick from 5: double angle from a given ratio, from a given ratio, from one ratio and the quadrant, exact values of half a special angle and reciprocal functions, given tan or cot, roots in the ratio. Tick as many as you want and set how many of each, or let it spread them evenly.

Doubling the ratio instead of the angle. sin 2θ is not 2 sin θ, and the gap is large: with sin θ = 3/5 the two give 24/25 and 6/5, and the second is not even possible.

Double angle identities worksheet for Precalculus

Select your difficulty

Use the double-angle identities

Find exact values with half angles

Use the double-angle identities

Find exact values with half angles

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these double angle identities worksheets free?+

Do the double angle identities worksheets come with an answer key?+

Are these double angle identities worksheets printable?+

What grade are these double angle identities worksheets for?+

Can I get a different version of the same double angle identities worksheet?+

Is this a free alternative to Kuta Software for double angle identities?+

Can I choose how hard the double angle identities questions are?+

What kinds of question are on a double angle identities worksheet?+

What do students get wrong with double angle identities?+