Solve the triangle: a = 24, c = 7, m∠ B = 83°

m∠ A is about 80.3°, m∠ C is about 16.7°, b is about 24.2

Law of cosines worksheet for Precalculus

Precalculus uses the law of cosines both ways: a side from two sides and the angle between them, and an angle from three sides. It is often the first step in solving a whole triangle before switching to the law of sines, with obtuse angles, exact radicals and answers to the nearest tenth.

Also taught in Algebra 2 and Geometry, each 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 The Law of Cosines Date Period Find the missing measure. Use the law of cosines to find the third side exactly In a triangle, $b = 43$ , $c = 41$ and $A = 90^{\circ}$ . Find $a$ exactly. In a triangle, $a = 64$ , $b = 53$ and $C = 120^{\circ}$ . Find $c$ exactly. In a triangle, $c = 62$ , $a = 68$ and $B = 120^{\circ}$ . Find $b$ exactly. In a triangle, $b = 38$ , $c = 65$ and $A = 60^{\circ}$ . Find $a$ exactly. In a triangle, $c = 40$ , $a = 67$ and $B = 90^{\circ}$ . Find $b$ exactly. In a triangle, $b = 44$ , $c = 70$ and $A = 90^{\circ}$ . Find $a$ exactly. In a triangle, $a = 33$ , $b = 64$ and $C = 120^{\circ}$ . Find $c$ exactly. Find the exact angle from three sides In a triangle, $b = 16$ , $c = 63$ and $a = 65$ . Find $A$ . In a triangle, $c = 16$ , $a = 55$ and $b = 49$ . Find $B$ . In a triangle, $a = 17$ , $b = 63$ and $c = 73$ . Find $C$ . In a triangle, $a = 33$ , $b = 56$ and $c = 65$ . Find $C$ . In a triangle, $a = 35$ , $b = 48$ and $c = 43$ . Find $C$ . In a triangle, $b = 32$ , $c = 45$ and $a = 67$ . Find $A$ . In a triangle, $b = 11$ , $c = 60$ and $a = 61$ . Find $A$ . Find a side or an angle with the law of cosines, to the nearest tenth In a triangle, $b = 40$ , $c = 9$ and $a = 35$ . Find $A$ to the nearest tenth of a degree. In a triangle, $c = 10$ , $a = 43$ and $b = 46$ . Find $B$ to the nearest tenth of a degree. In a triangle, $b = 20$ , $c = 23$ and $a = 42$ . Find $A$ to the nearest tenth of a degree. In a triangle, $b = 42$ , $c = 13$ and $a = 52$ . Find $A$ to the nearest tenth of a degree. In a triangle, $c = 4$ , $a = 19$ and $b = 20$ . Find $B$ to the nearest tenth of a degree. In a triangle, $c = 14$ , $a = 22$ and $b = 24$ . Find $B$ to the nearest tenth of a degree. Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Law of Cosines

Date Period

Find the missing measure.

Use the law of cosines to find the third side exactly
In a triangle, $b = 43$ , $c = 41$ and $A = 90^{\circ}$ . Find $a$ exactly.
In a triangle, $a = 64$ , $b = 53$ and $C = 120^{\circ}$ . Find $c$ exactly.
In a triangle, $c = 62$ , $a = 68$ and $B = 120^{\circ}$ . Find $b$ exactly.
In a triangle, $b = 38$ , $c = 65$ and $A = 60^{\circ}$ . Find $a$ exactly.
In a triangle, $c = 40$ , $a = 67$ and $B = 90^{\circ}$ . Find $b$ exactly.
In a triangle, $b = 44$ , $c = 70$ and $A = 90^{\circ}$ . Find $a$ exactly.
In a triangle, $a = 33$ , $b = 64$ and $C = 120^{\circ}$ . Find $c$ exactly.

Find the exact angle from three sides
In a triangle, $b = 16$ , $c = 63$ and $a = 65$ . Find $A$ .
In a triangle, $c = 16$ , $a = 55$ and $b = 49$ . Find $B$ .
In a triangle, $a = 17$ , $b = 63$ and $c = 73$ . Find $C$ .
In a triangle, $a = 33$ , $b = 56$ and $c = 65$ . Find $C$ .
In a triangle, $a = 35$ , $b = 48$ and $c = 43$ . Find $C$ .
In a triangle, $b = 32$ , $c = 45$ and $a = 67$ . Find $A$ .
In a triangle, $b = 11$ , $c = 60$ and $a = 61$ . Find $A$ .

Find a side or an angle with the law of cosines, to the nearest tenth
In a triangle, $b = 40$ , $c = 9$ and $a = 35$ . Find $A$ to the nearest tenth of a degree.
In a triangle, $c = 10$ , $a = 43$ and $b = 46$ . Find $B$ to the nearest tenth of a degree.
In a triangle, $b = 20$ , $c = 23$ and $a = 42$ . Find $A$ to the nearest tenth of a degree.
In a triangle, $b = 42$ , $c = 13$ and $a = 52$ . Find $A$ to the nearest tenth of a degree.
In a triangle, $c = 4$ , $a = 19$ and $b = 20$ . Find $B$ to the nearest tenth of a degree.
In a triangle, $c = 14$ , $a = 22$ and $b = 24$ . Find $B$ to the nearest tenth of a degree.



Sigma Prep · sigmaprep.io/worksheets Name The Law of CosinesAnswer keyVersion 2 Date Period Find the missing measure. Use the law of cosines to find the third side exactly In a triangle, $b = 43$ , $c = 41$ and $A = 90^{\circ}$ . Find $a$ exactly. $\sqrt{3530}$ In a triangle, $a = 64$ , $b = 53$ and $C = 120^{\circ}$ . Find $c$ exactly. $\sqrt{10297}$ In a triangle, $c = 62$ , $a = 68$ and $B = 120^{\circ}$ . Find $b$ exactly. $2 \sqrt{3171}$ In a triangle, $b = 38$ , $c = 65$ and $A = 60^{\circ}$ . Find $a$ exactly. $\sqrt{3199}$ In a triangle, $c = 40$ , $a = 67$ and $B = 90^{\circ}$ . Find $b$ exactly. $\sqrt{6089}$ In a triangle, $b = 44$ , $c = 70$ and $A = 90^{\circ}$ . Find $a$ exactly. $2 \sqrt{1709}$ In a triangle, $a = 33$ , $b = 64$ and $C = 120^{\circ}$ . Find $c$ exactly. $\sqrt{7297}$ Find the exact angle from three sides In a triangle, $b = 16$ , $c = 63$ and $a = 65$ . Find $A$ . $90^{\circ}$ In a triangle, $c = 16$ , $a = 55$ and $b = 49$ . Find $B$ . $60^{\circ}$ In a triangle, $a = 17$ , $b = 63$ and $c = 73$ . Find $C$ . $120^{\circ}$ In a triangle, $a = 33$ , $b = 56$ and $c = 65$ . Find $C$ . $90^{\circ}$ In a triangle, $a = 35$ , $b = 48$ and $c = 43$ . Find $C$ . $60^{\circ}$ In a triangle, $b = 32$ , $c = 45$ and $a = 67$ . Find $A$ . $120^{\circ}$ In a triangle, $b = 11$ , $c = 60$ and $a = 61$ . Find $A$ . $90^{\circ}$ Find a side or an angle with the law of cosines, to the nearest tenth In a triangle, $b = 40$ , $c = 9$ and $a = 35$ . Find $A$ to the nearest tenth of a degree. ${50.7}^{\circ}$ In a triangle, $c = 10$ , $a = 43$ and $b = 46$ . Find $B$ to the nearest tenth of a degree. ${101.2}^{\circ}$ In a triangle, $b = 20$ , $c = 23$ and $a = 42$ . Find $A$ to the nearest tenth of a degree. ${155.2}^{\circ}$ In a triangle, $b = 42$ , $c = 13$ and $a = 52$ . Find $A$ to the nearest tenth of a degree. ${134.9}^{\circ}$ In a triangle, $c = 4$ , $a = 19$ and $b = 20$ . Find $B$ to the nearest tenth of a degree. ${98.7}^{\circ}$ In a triangle, $c = 14$ , $a = 22$ and $b = 24$ . Find $B$ to the nearest tenth of a degree. ${80.3}^{\circ}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Law of CosinesAnswer keyVersion 2

Date Period

Find the missing measure.

Use the law of cosines to find the third side exactly
In a triangle, $b = 43$ , $c = 41$ and $A = 90^{\circ}$ . Find $a$ exactly. $\sqrt{3530}$
In a triangle, $a = 64$ , $b = 53$ and $C = 120^{\circ}$ . Find $c$ exactly. $\sqrt{10297}$
In a triangle, $c = 62$ , $a = 68$ and $B = 120^{\circ}$ . Find $b$ exactly. $2 \sqrt{3171}$
In a triangle, $b = 38$ , $c = 65$ and $A = 60^{\circ}$ . Find $a$ exactly. $\sqrt{3199}$
In a triangle, $c = 40$ , $a = 67$ and $B = 90^{\circ}$ . Find $b$ exactly. $\sqrt{6089}$
In a triangle, $b = 44$ , $c = 70$ and $A = 90^{\circ}$ . Find $a$ exactly. $2 \sqrt{1709}$
In a triangle, $a = 33$ , $b = 64$ and $C = 120^{\circ}$ . Find $c$ exactly. $\sqrt{7297}$

Find the exact angle from three sides
In a triangle, $b = 16$ , $c = 63$ and $a = 65$ . Find $A$ . $90^{\circ}$
In a triangle, $c = 16$ , $a = 55$ and $b = 49$ . Find $B$ . $60^{\circ}$
In a triangle, $a = 17$ , $b = 63$ and $c = 73$ . Find $C$ . $120^{\circ}$
In a triangle, $a = 33$ , $b = 56$ and $c = 65$ . Find $C$ . $90^{\circ}$
In a triangle, $a = 35$ , $b = 48$ and $c = 43$ . Find $C$ . $60^{\circ}$
In a triangle, $b = 32$ , $c = 45$ and $a = 67$ . Find $A$ . $120^{\circ}$
In a triangle, $b = 11$ , $c = 60$ and $a = 61$ . Find $A$ . $90^{\circ}$

Find a side or an angle with the law of cosines, to the nearest tenth
In a triangle, $b = 40$ , $c = 9$ and $a = 35$ . Find $A$ to the nearest tenth of a degree. ${50.7}^{\circ}$
In a triangle, $c = 10$ , $a = 43$ and $b = 46$ . Find $B$ to the nearest tenth of a degree. ${101.2}^{\circ}$
In a triangle, $b = 20$ , $c = 23$ and $a = 42$ . Find $A$ to the nearest tenth of a degree. ${155.2}^{\circ}$
In a triangle, $b = 42$ , $c = 13$ and $a = 52$ . Find $A$ to the nearest tenth of a degree. ${134.9}^{\circ}$
In a triangle, $c = 4$ , $a = 19$ and $b = 20$ . Find $B$ to the nearest tenth of a degree. ${98.7}^{\circ}$
In a triangle, $c = 14$ , $a = 22$ and $b = 24$ . Find $B$ to the nearest tenth of a degree. ${80.3}^{\circ}$

How to do these

In a triangle, $a = 9$ , $b = 14$ and $c = 20$ . Find $C$ to the nearest tenth of a degree.

cos C = (a² + b² - c²) ÷ 2ab = (81 + 196 - 400) ÷ 252.
That is -123/252, about -0.488, and inverse cosine gives C ≈ 119.2°.

The answer is ${119.2}^{\circ}$ .

Where students go wrong

Rearranging with the wrong sign. cos C is (a² + b² - c²) over 2ab; writing c² - a² - b² on top turns this obtuse angle into an acute one.

Also called cosine rule or solving triangles.

What you can put on this worksheet

A missing side, from two sides and the angle between: In a triangle, $a = 5$ , $b = 7$ and $C = 120^{\circ}$ . Find $c$ exactly. → $\sqrt{109}$
A missing angle, from three sides: In a triangle, $c = 7$ , $a = 15$ and $b = 13$ . Find $B$ . → $60^{\circ}$
The law of cosines, any angle, rounded: In a triangle, $b = 13$ , $c = 8$ and $A = 146^{\circ}$ . Find $a$ to the nearest tenth. → $20.1$
Solve the whole triangle: Solve the triangle:
$a = 24, c = 7, m ∠ B = 83^{\circ}$ → $m ∠ A \approx {80.3}^{\circ},$ $m ∠ C \approx {16.7}^{\circ},$ $b \approx 24.2$
From a labeled drawing: Find $m ∠ Y$ to the nearest tenth of a degree. — triangle xyz, labeled Y, Z, X, 5, 11, 10 → $m ∠ Y \approx {65.3}^{\circ}$

Questions about these worksheets

Yes. Every law of cosines 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 law of cosines 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 two 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: a missing side, from two sides and the angle between, a missing angle, from three sides, the law of cosines, any angle, rounded, solve the whole triangle and from a labeled drawing. Tick as many as you want and set how many of each, or let it spread them evenly.

Adding the correction term instead of subtracting it, or using an angle that is not between the two sides. The angle has to sit between them or the rule does not apply.

Law of cosines worksheet for Precalculus

Select your difficulty

Use the law of cosines to find the third side exactly

Find the exact angle from three sides

Find a side or an angle with the law of cosines, to the nearest tenth

Use the law of cosines to find the third side exactly

Find the exact angle from three sides

Find a side or an angle with the law of cosines, to the nearest tenth

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these law of cosines worksheets free?+

Do the law of cosines worksheets come with an answer key?+

Are these law of cosines worksheets printable?+

What grade are these law of cosines worksheets for?+

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Is this a free alternative to Kuta Software for law of cosines?+

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What do students get wrong with law of cosines?+