Half angle identities worksheet

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.



Sigma Prep · sigmaprep.io/worksheets Name Half‑Angle Identities Date Period Find each value exactly. If $\cos θ = \frac{35}{37}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ If $\cos θ = \frac{28}{53}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ If $\cos θ = \frac{12}{37}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ If $\cos θ = \frac{45}{53}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ If $\cos θ = \frac{21}{29}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ If $\cos θ = \frac{9}{41}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ Find $\tan 10 5^{\circ}$ exactly If $\cos θ = \frac{21}{29}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ If $\cos θ = \frac{45}{53}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ If $\cos θ = \frac{12}{37}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ If $\cos θ = \frac{20}{29}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ If $\cos θ = \frac{40}{41}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ If $\cos θ = \frac{12}{37}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ Find $\tan 112. 5^{\circ}$ exactly If $\cos θ = \frac{20}{29}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ If $\cos θ = \frac{40}{41}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ If $\cos θ = \frac{28}{53}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ If $\cos θ = \frac{21}{29}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ If $\cos θ = \frac{9}{41}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ If $\cos θ = \frac{20}{29}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ 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.

If $\cos θ = \frac{35}{37}$ and $θ$ is acute, find $\sin \frac{θ}{2}$
If $\cos θ = \frac{28}{53}$ and $θ$ is acute, find $\tan \frac{θ}{2}$
If $\cos θ = \frac{12}{37}$ and $θ$ is acute, find $\sin \frac{θ}{2}$
If $\cos θ = \frac{45}{53}$ and $θ$ is acute, find $\sin \frac{θ}{2}$
If $\cos θ = \frac{21}{29}$ and $θ$ is acute, find $\tan \frac{θ}{2}$
If $\cos θ = \frac{9}{41}$ and $θ$ is acute, find $\tan \frac{θ}{2}$
Find $\tan 10 5^{\circ}$ exactly
If $\cos θ = \frac{21}{29}$ and $θ$ is acute, find $\sin \frac{θ}{2}$
If $\cos θ = \frac{45}{53}$ and $θ$ is acute, find $\tan \frac{θ}{2}$
If $\cos θ = \frac{12}{37}$ and $θ$ is acute, find $\cos \frac{θ}{2}$
If $\cos θ = \frac{20}{29}$ and $θ$ is acute, find $\sin \frac{θ}{2}$
If $\cos θ = \frac{40}{41}$ and $θ$ is acute, find $\tan \frac{θ}{2}$
If $\cos θ = \frac{12}{37}$ and $θ$ is acute, find $\tan \frac{θ}{2}$
Find $\tan 112. 5^{\circ}$ exactly
If $\cos θ = \frac{20}{29}$ and $θ$ is acute, find $\cos \frac{θ}{2}$
If $\cos θ = \frac{40}{41}$ and $θ$ is acute, find $\cos \frac{θ}{2}$
If $\cos θ = \frac{28}{53}$ and $θ$ is acute, find $\cos \frac{θ}{2}$
If $\cos θ = \frac{21}{29}$ and $θ$ is acute, find $\cos \frac{θ}{2}$
If $\cos θ = \frac{9}{41}$ and $θ$ is acute, find $\sin \frac{θ}{2}$
If $\cos θ = \frac{20}{29}$ and $θ$ is acute, find $\tan \frac{θ}{2}$



Sigma Prep · sigmaprep.io/worksheets Name Half‑Angle IdentitiesAnswer keyVersion 1 Date Period Find each value exactly. If $\cos θ = \frac{35}{37}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ $\frac{\sqrt{37}}{37}$ If $\cos θ = \frac{28}{53}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ $\frac{5}{9}$ If $\cos θ = \frac{12}{37}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ $\frac{5 \sqrt{74}}{74}$ If $\cos θ = \frac{45}{53}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ $\frac{2 \sqrt{53}}{53}$ If $\cos θ = \frac{21}{29}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ $\frac{2}{5}$ If $\cos θ = \frac{9}{41}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ $\frac{4}{5}$ Find $\tan 10 5^{\circ}$ exactly $- 2 - \sqrt{3}$ If $\cos θ = \frac{21}{29}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ $\frac{2 \sqrt{29}}{29}$ If $\cos θ = \frac{45}{53}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ $\frac{2}{7}$ If $\cos θ = \frac{12}{37}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ $\frac{7 \sqrt{74}}{74}$ If $\cos θ = \frac{20}{29}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ $\frac{3 \sqrt{58}}{58}$ If $\cos θ = \frac{40}{41}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ $\frac{1}{9}$ If $\cos θ = \frac{12}{37}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ $\frac{5}{7}$ Find $\tan 112. 5^{\circ}$ exactly $- \sqrt{2} - 1$ If $\cos θ = \frac{20}{29}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ $\frac{7 \sqrt{58}}{58}$ If $\cos θ = \frac{40}{41}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ $\frac{9 \sqrt{82}}{82}$ If $\cos θ = \frac{28}{53}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ $\frac{9 \sqrt{106}}{106}$ If $\cos θ = \frac{21}{29}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ $\frac{5 \sqrt{29}}{29}$ If $\cos θ = \frac{9}{41}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ $\frac{4 \sqrt{41}}{41}$ If $\cos θ = \frac{20}{29}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ $\frac{3}{7}$ 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.

If $\cos θ = \frac{35}{37}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ $\frac{\sqrt{37}}{37}$
If $\cos θ = \frac{28}{53}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ $\frac{5}{9}$
If $\cos θ = \frac{12}{37}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ $\frac{5 \sqrt{74}}{74}$
If $\cos θ = \frac{45}{53}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ $\frac{2 \sqrt{53}}{53}$
If $\cos θ = \frac{21}{29}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ $\frac{2}{5}$
If $\cos θ = \frac{9}{41}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ $\frac{4}{5}$
Find $\tan 10 5^{\circ}$ exactly $- 2 - \sqrt{3}$
If $\cos θ = \frac{21}{29}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ $\frac{2 \sqrt{29}}{29}$
If $\cos θ = \frac{45}{53}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ $\frac{2}{7}$
If $\cos θ = \frac{12}{37}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ $\frac{7 \sqrt{74}}{74}$
If $\cos θ = \frac{20}{29}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ $\frac{3 \sqrt{58}}{58}$
If $\cos θ = \frac{40}{41}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ $\frac{1}{9}$
If $\cos θ = \frac{12}{37}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ $\frac{5}{7}$
Find $\tan 112. 5^{\circ}$ exactly $- \sqrt{2} - 1$
If $\cos θ = \frac{20}{29}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ $\frac{7 \sqrt{58}}{58}$
If $\cos θ = \frac{40}{41}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ $\frac{9 \sqrt{82}}{82}$
If $\cos θ = \frac{28}{53}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ $\frac{9 \sqrt{106}}{106}$
If $\cos θ = \frac{21}{29}$ and $θ$ is acute, find $\cos \frac{θ}{2}$ $\frac{5 \sqrt{29}}{29}$
If $\cos θ = \frac{9}{41}$ and $θ$ is acute, find $\sin \frac{θ}{2}$ $\frac{4 \sqrt{41}}{41}$
If $\cos θ = \frac{20}{29}$ and $θ$ is acute, find $\tan \frac{θ}{2}$ $\frac{3}{7}$

How to do these

Find $\tan 1 5^{\circ}$ 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 $\tan 6 0^{\circ}$ exactly → $\sqrt{3}$

Questions about these worksheets

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.

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 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.

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 2: from a given ratio and exact values of half a special angle. Tick as many as you want and set how many of each, or let it spread them evenly.

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.

Half angle identities worksheet

Half‑Angle Identities

Half‑Angle IdentitiesAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these half angle identities worksheets free?+

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

Are these half angle identities worksheets printable?+

What grade are these half angle identities worksheets for?+

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

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

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

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

What do students get wrong with half angle identities?+