Half angle identities worksheet for Precalculus

Q: Are these half angle identities worksheets free?

Yes. Every half 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.

Q: Do the half angle identities worksheets come with an answer key?

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.

Q: Are these half angle identities worksheets printable?

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.

Q: What grade are these half angle identities worksheets for?

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.

Q: Can I get a different version of the same half angle identities worksheet?

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.

Q: Is this a free alternative to Kuta Software for half angle identities?

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 half 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.

Q: Can I choose how hard the half angle identities questions are?

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.

Q: What kinds of question are on a half angle identities worksheet?

You pick from 4: from a given ratio, exact values of half a special angle, outside the first quadrant, choosing the sign and power reducing. Tick as many as you want and set how many of each, or let it spread them evenly.

Q: What do students get wrong with half angle identities?

Halving the value instead of the angle. Sine of half an angle is not half the sine, and the whole reason these formulas exist is that it is not.

Halving an angle turns a value you know into one you did not, which is where the exact value of something like 15 degrees comes from. The tangent form is the tidiest: one minus the cosine, over the sine.

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

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



Sigma Prep · sigmaprep.io/worksheets Name Half-Angle Identities Date Period Find each value exactly, or rewrite as asked. Use the half-angle identities If $\cos θ = \frac{15}{17}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ If $\cos θ = \frac{4}{5}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ If $\cos θ = \frac{24}{25}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ If $\cos θ = \frac{15}{17}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ If $\cos θ = \frac{8}{17}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ If $\cos θ = \frac{7}{25}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ Use half-angle formulas, choosing the sign $\sin θ = \frac{7}{25}$ and $\frac{π}{2} < θ < π$ . Find $\tan \frac{θ}{2}$ $\cos θ = - \frac{3}{5}$ and $\frac{π}{2} < θ < π$ . Find $\tan \frac{θ}{2}$ $\sin θ = - \frac{3}{5}$ and $π < θ < \frac{3 π}{2}$ . Find $\sin \frac{θ}{2}$ $\cos θ = - \frac{3}{5}$ and $\frac{π}{2} < θ < π$ . Find $\sin \frac{θ}{2}$ $\sin θ = - \frac{12}{13}$ and $\frac{3 π}{2} < θ < 2 π$ . Find $\cos \frac{θ}{2}$ $\cos θ = - \frac{3}{5}$ and $π < θ < \frac{3 π}{2}$ . Find $\cos \frac{θ}{2}$ Rewrite using power-reducing formulas Rewrite with no power greater than 1: $\cos^{4} x$ Rewrite with no power greater than 1: $\sin^{2} x \cos^{2} x$ Rewrite with no power greater than 1: $\sin^{4} x$ Rewrite with no power greater than 1: $\sin^{2} (3 x)$ Rewrite with no power greater than 1: $\cos^{2} (2 x)$ Rewrite with no power greater than 1: $\cos^{2} x$ Find exact values with half angles Find $\tan 165^{\circ}$ exactly Find $\sin 15^{\circ}$ exactly Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Half-Angle Identities

Date Period

Find each value exactly, or rewrite as asked.

Use the half-angle identities
If $\cos θ = \frac{15}{17}$ and $θ$ is acute, find $\sin \frac{θ}{2}$
If $\cos θ = \frac{4}{5}$ and $θ$ is acute, find $\tan \frac{θ}{2}$
If $\cos θ = \frac{24}{25}$ and $θ$ is acute, find $\cos \frac{θ}{2}$
If $\cos θ = \frac{15}{17}$ and $θ$ is acute, find $\tan \frac{θ}{2}$
If $\cos θ = \frac{8}{17}$ and $θ$ is acute, find $\sin \frac{θ}{2}$
If $\cos θ = \frac{7}{25}$ and $θ$ is acute, find $\cos \frac{θ}{2}$

Use half-angle formulas, choosing the sign
$\sin θ = \frac{7}{25}$ and $\frac{π}{2} < θ < π$ . Find $\tan \frac{θ}{2}$
$\cos θ = - \frac{3}{5}$ and $\frac{π}{2} < θ < π$ . Find $\tan \frac{θ}{2}$
$\sin θ = - \frac{3}{5}$ and $π < θ < \frac{3 π}{2}$ . Find $\sin \frac{θ}{2}$
$\cos θ = - \frac{3}{5}$ and $\frac{π}{2} < θ < π$ . Find $\sin \frac{θ}{2}$
$\sin θ = - \frac{12}{13}$ and $\frac{3 π}{2} < θ < 2 π$ . Find $\cos \frac{θ}{2}$
$\cos θ = - \frac{3}{5}$ and $π < θ < \frac{3 π}{2}$ . Find $\cos \frac{θ}{2}$

Rewrite using power-reducing formulas
Rewrite with no power greater than 1: $\cos^{4} x$
Rewrite with no power greater than 1: $\sin^{2} x \cos^{2} x$
Rewrite with no power greater than 1: $\sin^{4} x$
Rewrite with no power greater than 1: $\sin^{2} (3 x)$
Rewrite with no power greater than 1: $\cos^{2} (2 x)$
Rewrite with no power greater than 1: $\cos^{2} x$

Find exact values with half angles
Find $\tan 165^{\circ}$ exactly
Find $\sin 15^{\circ}$ exactly



Sigma Prep · sigmaprep.io/worksheets Name Half-Angle IdentitiesAnswer keyVersion 1 Date Period Find each value exactly, or rewrite as asked. Use the half-angle identities If $\cos θ = \frac{15}{17}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ $\frac{\sqrt{17}}{17}$ If $\cos θ = \frac{4}{5}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ $\frac{1}{3}$ If $\cos θ = \frac{24}{25}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ $\frac{7 \sqrt{2}}{10}$ If $\cos θ = \frac{15}{17}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ $\frac{1}{4}$ If $\cos θ = \frac{8}{17}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ $\frac{3 \sqrt{34}}{34}$ If $\cos θ = \frac{7}{25}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ $\frac{4}{5}$ Use half-angle formulas, choosing the sign $\sin θ = \frac{7}{25}$ and $\frac{π}{2} < θ < π$ . Find $\tan \frac{θ}{2}$ $7$ $\cos θ = - \frac{3}{5}$ and $\frac{π}{2} < θ < π$ . Find $\tan \frac{θ}{2}$ $2$ $\sin θ = - \frac{3}{5}$ and $π < θ < \frac{3 π}{2}$ . Find $\sin \frac{θ}{2}$ $\frac{3 \sqrt{10}}{10}$ $\cos θ = - \frac{3}{5}$ and $\frac{π}{2} < θ < π$ . Find $\sin \frac{θ}{2}$ $\frac{2 \sqrt{5}}{5}$ $\sin θ = - \frac{12}{13}$ and $\frac{3 π}{2} < θ < 2 π$ . Find $\cos \frac{θ}{2}$ $- \frac{3 \sqrt{13}}{13}$ $\cos θ = - \frac{3}{5}$ and $π < θ < \frac{3 π}{2}$ . Find $\cos \frac{θ}{2}$ $- \frac{\sqrt{5}}{5}$ Rewrite using power-reducing formulas Rewrite with no power greater than 1: $\cos^{4} x$ $\frac{3}{8} + \frac{1}{2} \cos 2 x + \frac{1}{8} \cos 4 x$ Rewrite with no power greater than 1: $\sin^{2} x \cos^{2} x$ $\frac{1}{8} - \frac{1}{8} \cos 4 x$ Rewrite with no power greater than 1: $\sin^{4} x$ $\frac{3}{8} - \frac{1}{2} \cos 2 x + \frac{1}{8} \cos 4 x$ Rewrite with no power greater than 1: $\sin^{2} (3 x)$ $\frac{1}{2} - \frac{1}{2} \cos 6 x$ Rewrite with no power greater than 1: $\cos^{2} (2 x)$ $\frac{1}{2} + \frac{1}{2} \cos 4 x$ Rewrite with no power greater than 1: $\cos^{2} x$ $\frac{1}{2} + \frac{1}{2} \cos 2 x$ Find exact values with half angles Find $\tan 165^{\circ}$ exactly $\sqrt{3} - 2$ Find $\sin 15^{\circ}$ exactly $\frac{\sqrt{6} - \sqrt{2}}{4}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Half-Angle IdentitiesAnswer keyVersion 1

Date Period

Find each value exactly, or rewrite as asked.

Use the half-angle identities
If $\cos θ = \frac{15}{17}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ $\frac{\sqrt{17}}{17}$
If $\cos θ = \frac{4}{5}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ $\frac{1}{3}$
If $\cos θ = \frac{24}{25}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ $\frac{7 \sqrt{2}}{10}$
If $\cos θ = \frac{15}{17}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ $\frac{1}{4}$
If $\cos θ = \frac{8}{17}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ $\frac{3 \sqrt{34}}{34}$
If $\cos θ = \frac{7}{25}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ $\frac{4}{5}$

Use half-angle formulas, choosing the sign
$\sin θ = \frac{7}{25}$ and $\frac{π}{2} < θ < π$ . Find $\tan \frac{θ}{2}$ $7$
$\cos θ = - \frac{3}{5}$ and $\frac{π}{2} < θ < π$ . Find $\tan \frac{θ}{2}$ $2$
$\sin θ = - \frac{3}{5}$ and $π < θ < \frac{3 π}{2}$ . Find $\sin \frac{θ}{2}$ $\frac{3 \sqrt{10}}{10}$
$\cos θ = - \frac{3}{5}$ and $\frac{π}{2} < θ < π$ . Find $\sin \frac{θ}{2}$ $\frac{2 \sqrt{5}}{5}$
$\sin θ = - \frac{12}{13}$ and $\frac{3 π}{2} < θ < 2 π$ . Find $\cos \frac{θ}{2}$ $- \frac{3 \sqrt{13}}{13}$
$\cos θ = - \frac{3}{5}$ and $π < θ < \frac{3 π}{2}$ . Find $\cos \frac{θ}{2}$ $- \frac{\sqrt{5}}{5}$

Rewrite using power-reducing formulas
Rewrite with no power greater than 1: $\cos^{4} x$ $\frac{3}{8} + \frac{1}{2} \cos 2 x + \frac{1}{8} \cos 4 x$
Rewrite with no power greater than 1: $\sin^{2} x \cos^{2} x$ $\frac{1}{8} - \frac{1}{8} \cos 4 x$
Rewrite with no power greater than 1: $\sin^{4} x$ $\frac{3}{8} - \frac{1}{2} \cos 2 x + \frac{1}{8} \cos 4 x$
Rewrite with no power greater than 1: $\sin^{2} (3 x)$ $\frac{1}{2} - \frac{1}{2} \cos 6 x$
Rewrite with no power greater than 1: $\cos^{2} (2 x)$ $\frac{1}{2} + \frac{1}{2} \cos 4 x$
Rewrite with no power greater than 1: $\cos^{2} x$ $\frac{1}{2} + \frac{1}{2} \cos 2 x$

Find exact values with half angles
Find $\tan 165^{\circ}$ exactly $\sqrt{3} - 2$
Find $\sin 15^{\circ}$ exactly $\frac{\sqrt{6} - \sqrt{2}}{4}$

How to do these

Find $\tan 15^{\circ}$ exactly

15 is half of 30.
One minus cos 30 is 1 minus root 3 over 2.
Divide that by sin 30, which is a half.

The answer is $2 - \sqrt{3}$ .

Where students go wrong

Halving the value instead of the angle. Sine of half an angle is not half the sine, and the whole reason these formulas exist is that it is not.

Also called half angle formulas, exact value of tan 15 or reduction identities.

What you can put on this worksheet

From a given ratio: If $\cos θ = \frac{8}{17}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ → $\frac{5 \sqrt{34}}{34}$
Exact values of half a special angle: Find $\sin 75^{\circ}$ exactly → $\frac{\sqrt{6} + \sqrt{2}}{4}$
Outside the first quadrant, choosing the sign: $\cos θ = \frac{15}{17}$ and $\frac{3 π}{2} < θ < 2 π$ . Find $\cos \frac{θ}{2}$ → $- \frac{4 \sqrt{17}}{17}$
Power reducing: Rewrite with no power greater than 1: $\cos^{2} (2 x)$ → $\frac{1}{2} + \frac{1}{2} \cos 4 x$

Questions about these worksheets