Solve the triangle: m∠ B = 28°, b = 29, c = 24, to the nearest tenth.

m∠ A is about 129.1°, m∠ C is about 22.9°, a is about 47.9

Law of sines worksheet for Precalculus

Precalculus adds the ambiguous case to the law of sines: with two sides and an angle not between them there may be no triangle, one, or two. Every fourth question asks how many triangles are possible. The rest find a missing side, exact or to the nearest tenth, and one finds a missing angle.

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 Sines Date Period Find each value asked for. Use the law of sines In $△ A B C$ , $∠ B = 120^{\circ}$ and $∠ C = 45^{\circ}$ . The side opposite $B$ is $4$ . How long is the side opposite $C$ ? Give an exact answer. In $△ A B C$ , $m ∠ A = 67^{\circ}$ , $m ∠ C = 55^{\circ}$ and $a = 16$ . Use the law of sines to find $c$ , rounded to the nearest tenth. In a triangle, $A = 45^{\circ}$ , $C = 120^{\circ}$ and $a = 2$ . Find $c$ exactly. In $△ A B C$ , $c = 27$ , $m ∠ B = 54^{\circ}$ and $m ∠ C = 66^{\circ}$ . Find $b$ to the nearest tenth. A triangle has a $135^{\circ}$ angle facing a side of $7$ . Another of its angles is $30^{\circ}$ . Find the exact length of the side facing the $30^{\circ}$ angle. $△ A B C$ has $m ∠ A = 34^{\circ}$ , $m ∠ B = 55^{\circ}$ and $b = 27$ . How long is side $a$ , to the nearest tenth? In $△ A B C$ , $∠ A = 120^{\circ}$ and $∠ B = 30^{\circ}$ . The side opposite $A$ is $4$ . How long is the side opposite $B$ ? Give an exact answer. $△ A B C$ has $m ∠ A = 52^{\circ}$ , $m ∠ C = 55^{\circ}$ and $c = 20$ . Find $a$ to the nearest tenth. In a triangle, $C = 45^{\circ}$ , $B = 60^{\circ}$ and $c = 4$ . Find $b$ exactly. In $△ A B C$ , $m ∠ B = 76^{\circ}$ , $m ∠ C = 48^{\circ}$ and $b = 17$ . How long is side $c$ , to the nearest tenth? In $△ A B C$ , $∠ B = 45^{\circ}$ and $∠ A = 30^{\circ}$ . The side opposite $B$ is $5$ . How long is the side opposite $A$ ? Give an exact answer. In $△ A B C$ , $a = 16$ , $m ∠ A = 78^{\circ}$ and $m ∠ B = 28^{\circ}$ . Use the law of sines to find $b$ , rounded to the nearest tenth. A triangle has a $30^{\circ}$ angle facing a side of $8$ . Another of its angles is $45^{\circ}$ . Find the exact length of the side facing the $45^{\circ}$ angle. In $△ A B C$ , $b = 18$ , $m ∠ B = 37^{\circ}$ and $m ∠ C = 88^{\circ}$ . How long is side $c$ , to the nearest tenth? Find how many triangles are possible In a triangle, $B = 52^{\circ}$ , $b = 6$ and $c = 7$ . How many triangles are possible? In a triangle, $C = 72^{\circ}$ , $a = 9$ and $c = 7$ . How many triangles are possible? In a triangle, $A = 31^{\circ}$ , $a = 16$ and $b = 6$ . How many triangles are possible? In a triangle, $C = 71^{\circ}$ , $a = 15$ and $c = 18$ . How many triangles are possible? In a triangle, $B = 38^{\circ}$ , $b = 5$ and $c = 7$ . How many triangles are possible? In a triangle, $A = 73^{\circ}$ , $a = 9$ and $b = 12$ . How many triangles are possible? Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Law of Sines

Date Period

Find each value asked for.

Use the law of sines
In $△ A B C$ , $∠ B = 120^{\circ}$ and $∠ C = 45^{\circ}$ . The side opposite $B$ is $4$ . How long is the side opposite $C$ ? Give an exact answer.
In $△ A B C$ , $m ∠ A = 67^{\circ}$ , $m ∠ C = 55^{\circ}$ and $a = 16$ . Use the law of sines to find $c$ , rounded to the nearest tenth.
In a triangle, $A = 45^{\circ}$ , $C = 120^{\circ}$ and $a = 2$ . Find $c$ exactly.
In $△ A B C$ , $c = 27$ , $m ∠ B = 54^{\circ}$ and $m ∠ C = 66^{\circ}$ . Find $b$ to the nearest tenth.
A triangle has a $135^{\circ}$ angle facing a side of $7$ . Another of its angles is $30^{\circ}$ . Find the exact length of the side facing the $30^{\circ}$ angle.
$△ A B C$ has $m ∠ A = 34^{\circ}$ , $m ∠ B = 55^{\circ}$ and $b = 27$ . How long is side $a$ , to the nearest tenth?
In $△ A B C$ , $∠ A = 120^{\circ}$ and $∠ B = 30^{\circ}$ . The side opposite $A$ is $4$ . How long is the side opposite $B$ ? Give an exact answer.
$△ A B C$ has $m ∠ A = 52^{\circ}$ , $m ∠ C = 55^{\circ}$ and $c = 20$ . Find $a$ to the nearest tenth.
In a triangle, $C = 45^{\circ}$ , $B = 60^{\circ}$ and $c = 4$ . Find $b$ exactly.
In $△ A B C$ , $m ∠ B = 76^{\circ}$ , $m ∠ C = 48^{\circ}$ and $b = 17$ . How long is side $c$ , to the nearest tenth?
In $△ A B C$ , $∠ B = 45^{\circ}$ and $∠ A = 30^{\circ}$ . The side opposite $B$ is $5$ . How long is the side opposite $A$ ? Give an exact answer.
In $△ A B C$ , $a = 16$ , $m ∠ A = 78^{\circ}$ and $m ∠ B = 28^{\circ}$ . Use the law of sines to find $b$ , rounded to the nearest tenth.
A triangle has a $30^{\circ}$ angle facing a side of $8$ . Another of its angles is $45^{\circ}$ . Find the exact length of the side facing the $45^{\circ}$ angle.
In $△ A B C$ , $b = 18$ , $m ∠ B = 37^{\circ}$ and $m ∠ C = 88^{\circ}$ . How long is side $c$ , to the nearest tenth?

Find how many triangles are possible
In a triangle, $B = 52^{\circ}$ , $b = 6$ and $c = 7$ . How many triangles are possible?
In a triangle, $C = 72^{\circ}$ , $a = 9$ and $c = 7$ . How many triangles are possible?
In a triangle, $A = 31^{\circ}$ , $a = 16$ and $b = 6$ . How many triangles are possible?
In a triangle, $C = 71^{\circ}$ , $a = 15$ and $c = 18$ . How many triangles are possible?
In a triangle, $B = 38^{\circ}$ , $b = 5$ and $c = 7$ . How many triangles are possible?
In a triangle, $A = 73^{\circ}$ , $a = 9$ and $b = 12$ . How many triangles are possible?



Sigma Prep · sigmaprep.io/worksheets Name The Law of SinesAnswer keyVersion 2 Date Period Find each value asked for. Use the law of sines In $△ A B C$ , $∠ B = 120^{\circ}$ and $∠ C = 45^{\circ}$ . The side opposite $B$ is $4$ . How long is the side opposite $C$ ? Give an exact answer. $\frac{4 \sqrt{6}}{3}$ In $△ A B C$ , $m ∠ A = 67^{\circ}$ , $m ∠ C = 55^{\circ}$ and $a = 16$ . Use the law of sines to find $c$ , rounded to the nearest tenth. $14.2$ In a triangle, $A = 45^{\circ}$ , $C = 120^{\circ}$ and $a = 2$ . Find $c$ exactly. $\sqrt{6}$ In $△ A B C$ , $c = 27$ , $m ∠ B = 54^{\circ}$ and $m ∠ C = 66^{\circ}$ . Find $b$ to the nearest tenth. $23.9$ A triangle has a $135^{\circ}$ angle facing a side of $7$ . Another of its angles is $30^{\circ}$ . Find the exact length of the side facing the $30^{\circ}$ angle. $\frac{7 \sqrt{2}}{2}$ $△ A B C$ has $m ∠ A = 34^{\circ}$ , $m ∠ B = 55^{\circ}$ and $b = 27$ . How long is side $a$ , to the nearest tenth? $18.4$ In $△ A B C$ , $∠ A = 120^{\circ}$ and $∠ B = 30^{\circ}$ . The side opposite $A$ is $4$ . How long is the side opposite $B$ ? Give an exact answer. $\frac{4 \sqrt{3}}{3}$ $△ A B C$ has $m ∠ A = 52^{\circ}$ , $m ∠ C = 55^{\circ}$ and $c = 20$ . Find $a$ to the nearest tenth. $19.2$ In a triangle, $C = 45^{\circ}$ , $B = 60^{\circ}$ and $c = 4$ . Find $b$ exactly. $2 \sqrt{6}$ In $△ A B C$ , $m ∠ B = 76^{\circ}$ , $m ∠ C = 48^{\circ}$ and $b = 17$ . How long is side $c$ , to the nearest tenth? $13.0$ In $△ A B C$ , $∠ B = 45^{\circ}$ and $∠ A = 30^{\circ}$ . The side opposite $B$ is $5$ . How long is the side opposite $A$ ? Give an exact answer. $\frac{5 \sqrt{2}}{2}$ In $△ A B C$ , $a = 16$ , $m ∠ A = 78^{\circ}$ and $m ∠ B = 28^{\circ}$ . Use the law of sines to find $b$ , rounded to the nearest tenth. $7.7$ A triangle has a $30^{\circ}$ angle facing a side of $8$ . Another of its angles is $45^{\circ}$ . Find the exact length of the side facing the $45^{\circ}$ angle. $8 \sqrt{2}$ In $△ A B C$ , $b = 18$ , $m ∠ B = 37^{\circ}$ and $m ∠ C = 88^{\circ}$ . How long is side $c$ , to the nearest tenth? $29.9$ Find how many triangles are possible In a triangle, $B = 52^{\circ}$ , $b = 6$ and $c = 7$ . How many triangles are possible? $2$ In a triangle, $C = 72^{\circ}$ , $a = 9$ and $c = 7$ . How many triangles are possible? $0$ In a triangle, $A = 31^{\circ}$ , $a = 16$ and $b = 6$ . How many triangles are possible? $1$ In a triangle, $C = 71^{\circ}$ , $a = 15$ and $c = 18$ . How many triangles are possible? $1$ In a triangle, $B = 38^{\circ}$ , $b = 5$ and $c = 7$ . How many triangles are possible? $2$ In a triangle, $A = 73^{\circ}$ , $a = 9$ and $b = 12$ . How many triangles are possible? $0$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Law of SinesAnswer keyVersion 2

Date Period

Find each value asked for.

Use the law of sines
In $△ A B C$ , $∠ B = 120^{\circ}$ and $∠ C = 45^{\circ}$ . The side opposite $B$ is $4$ . How long is the side opposite $C$ ? Give an exact answer. $\frac{4 \sqrt{6}}{3}$
In $△ A B C$ , $m ∠ A = 67^{\circ}$ , $m ∠ C = 55^{\circ}$ and $a = 16$ . Use the law of sines to find $c$ , rounded to the nearest tenth. $14.2$
In a triangle, $A = 45^{\circ}$ , $C = 120^{\circ}$ and $a = 2$ . Find $c$ exactly. $\sqrt{6}$
In $△ A B C$ , $c = 27$ , $m ∠ B = 54^{\circ}$ and $m ∠ C = 66^{\circ}$ . Find $b$ to the nearest tenth. $23.9$
A triangle has a $135^{\circ}$ angle facing a side of $7$ . Another of its angles is $30^{\circ}$ . Find the exact length of the side facing the $30^{\circ}$ angle. $\frac{7 \sqrt{2}}{2}$
$△ A B C$ has $m ∠ A = 34^{\circ}$ , $m ∠ B = 55^{\circ}$ and $b = 27$ . How long is side $a$ , to the nearest tenth? $18.4$
In $△ A B C$ , $∠ A = 120^{\circ}$ and $∠ B = 30^{\circ}$ . The side opposite $A$ is $4$ . How long is the side opposite $B$ ? Give an exact answer. $\frac{4 \sqrt{3}}{3}$
$△ A B C$ has $m ∠ A = 52^{\circ}$ , $m ∠ C = 55^{\circ}$ and $c = 20$ . Find $a$ to the nearest tenth. $19.2$
In a triangle, $C = 45^{\circ}$ , $B = 60^{\circ}$ and $c = 4$ . Find $b$ exactly. $2 \sqrt{6}$
In $△ A B C$ , $m ∠ B = 76^{\circ}$ , $m ∠ C = 48^{\circ}$ and $b = 17$ . How long is side $c$ , to the nearest tenth? $13.0$
In $△ A B C$ , $∠ B = 45^{\circ}$ and $∠ A = 30^{\circ}$ . The side opposite $B$ is $5$ . How long is the side opposite $A$ ? Give an exact answer. $\frac{5 \sqrt{2}}{2}$
In $△ A B C$ , $a = 16$ , $m ∠ A = 78^{\circ}$ and $m ∠ B = 28^{\circ}$ . Use the law of sines to find $b$ , rounded to the nearest tenth. $7.7$
A triangle has a $30^{\circ}$ angle facing a side of $8$ . Another of its angles is $45^{\circ}$ . Find the exact length of the side facing the $45^{\circ}$ angle. $8 \sqrt{2}$
In $△ A B C$ , $b = 18$ , $m ∠ B = 37^{\circ}$ and $m ∠ C = 88^{\circ}$ . How long is side $c$ , to the nearest tenth? $29.9$

Find how many triangles are possible
In a triangle, $B = 52^{\circ}$ , $b = 6$ and $c = 7$ . How many triangles are possible? $2$
In a triangle, $C = 72^{\circ}$ , $a = 9$ and $c = 7$ . How many triangles are possible? $0$
In a triangle, $A = 31^{\circ}$ , $a = 16$ and $b = 6$ . How many triangles are possible? $1$
In a triangle, $C = 71^{\circ}$ , $a = 15$ and $c = 18$ . How many triangles are possible? $1$
In a triangle, $B = 38^{\circ}$ , $b = 5$ and $c = 7$ . How many triangles are possible? $2$
In a triangle, $A = 73^{\circ}$ , $a = 9$ and $b = 12$ . How many triangles are possible? $0$

How to do these

In a triangle, $A = 30^{\circ}$ , $a = 6$ and $b = 10$ . How many triangles are possible?

sin B = 10 sin 30° ÷ 6 = 5/6, so one answer is B ≈ 56.4°.
The obtuse partner is 180° - 56.4° = 123.6°, and 30° + 123.6° is under 180°, so it also fits: two triangles.

The answer is $2$ .

Where students go wrong

Stopping at the calculator's angle. Inverse sine only returns an acute angle, so the obtuse partner 180° - B has to be checked every time, and here it gives a second triangle.

Also called sine rule or solving oblique triangles.

What you can put on this worksheet

The law of sines: In a triangle, $C = 60^{\circ}$ , $B = 45^{\circ}$ and $c = 6$ . Find $b$ exactly. → $2 \sqrt{6}$
The law of sines, any angle, rounded: In $△ A B C$ , $m ∠ A = 70^{\circ}$ , $m ∠ C = 73^{\circ}$ and $c = 11$ . How long is side $a$ , to the nearest tenth? → $10.8$
The ambiguous case: how many triangles?: In a triangle, $B = 51^{\circ}$ , $b = 14$ and $c = 13$ . How many triangles are possible? → $1$
The ambiguous case: every possible angle: In $△ A B C$ , $m ∠ C = 51^{\circ}$ , $a = 13$ and $c = 14$ . Find all possible measures of $∠ A$ , to the nearest tenth of a degree. → $m ∠ A \approx {46.2}^{\circ}$
Area from two sides and the angle between: In $△ A B C$ , $m ∠ B = 90^{\circ}$ , $B C = 7$ and $A B = 8$ . Find the exact area of the triangle. → $28$
Solve the whole triangle, including the two-triangle case: Solve the triangle:
$m ∠ B = 28^{\circ}, b = 29, c = 24$ , to the nearest tenth. → $m ∠ A \approx {129.1}^{\circ},$ $m ∠ C \approx {22.9}^{\circ},$ $a \approx 47.9$

Questions about these worksheets

Yes. Every law of sines 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 sines 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 6: the law of sines, the law of sines, any angle, rounded, the ambiguous case: how many triangles?, the ambiguous case: every possible angle, area from two sides and the angle between and solve the whole triangle, including the two-triangle case. Tick as many as you want and set how many of each, or let it spread them evenly.

Pairing a side with the wrong angle. The rule only works when the angle is opposite the side, not merely next to it.

Law of sines worksheet for Precalculus

Select your difficulty

Use the law of sines

Find how many triangles are possible

Use the law of sines

Find how many triangles are possible

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these law of sines worksheets free?+

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

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