A ship leaves port and sails 10 mi on a course of S 15° E, then turns 110° to the left and goes 60 mi. To the nearest tenth, give the ship's distance from port and the ship's bearing from port. Give the bearing as a three-figure bearing, measured clockwise from north (like 045.0°).

57.4 mi on a bearing of 064.4°

A boat leaves the harbor and travels 45 mi on a course of S 75° W, then turns 75° to the right and goes 25 mi. How far is the boat from the harbor, and on what bearing? Round both to the nearest tenth. Give the bearing as a three-figure bearing, measured clockwise from north (like 045.0°).

56.9 mi on a bearing of 280.1°

From the airport, the plane flies 115 mi on a course of S 35° W. The plane then turns 115° to the right and goes 60 mi more. How far from the airport does the plane end up, and in what direction? Give the distance and the bearing to the nearest tenth. Give the bearing as a three-figure bearing, measured clockwise from north (like 045.0°).

104.8 mi on a bearing of 246.2°

A hiker leaves camp and walks 9 mi on a bearing of 030°, then 11 mi on a bearing of 270°. What are the hiker's distance and bearing from camp now? Round both to the nearest tenth.

10.1 mi on a bearing of 320.2°

Worksheets · Precalculus

Law of sines and cosines worksheet for Precalculus

Precalculus uses both laws on any triangle: the law of sines when an angle and the side opposite it are known, and the law of cosines for two sides and the angle between them or for all three sides. Answers are exact values from special angles or calculator values rounded to the tenth, and bearing problems put the same work on a map.

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

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Sigma Prep · sigmaprep.io/worksheets Name The Laws of Sines and Cosines Date Period Find each measure asked for. Use the law of sines In $△ A B C$ , $∠ B = 120^{\circ}$ , the side opposite it is $36$ , and the side opposite $∠ C$ is $12 \sqrt{6}$ . Find $∠ A$ . In $△ A B C$ , $a = 35$ , $b = 21$ and $m ∠ A = 42^{\circ}$ . What is $m ∠ B$ , to the nearest tenth of a degree? In a triangle, $B = 45^{\circ}$ , $b = 28$ and $a = 14 \sqrt{2}$ . Find $C$ . $△ A B C$ has $m ∠ B = 27^{\circ}$ , $a = 16$ and $b = 21$ . Find $m ∠ A$ to the nearest tenth of a degree. A triangle has a $60^{\circ}$ angle facing a side of $42$ and a second side of $14 \sqrt{6}$ . Find the angle between those two sides. In $△ A B C$ , $m ∠ A = 75^{\circ}$ , $a = 15$ and $c = 5$ . Use the law of sines to find the measure of $∠ C$ , to the nearest tenth of a degree. In $△ A B C$ , $∠ A = 120^{\circ}$ , the side opposite it is $48$ , and the side opposite $∠ B$ is $16 \sqrt{3}$ . Find $∠ C$ . Use the law of cosines to find the third side exactly In a triangle, $b = 55$ , $c = 50$ and $A = 90^{\circ}$ . Find $a$ exactly. In a triangle, $a = 63$ , $b = 65$ and $C = 120^{\circ}$ . Find $c$ exactly. In a triangle, $b = 68$ , $c = 36$ and $A = 90^{\circ}$ . Find $a$ exactly. In a triangle, $b = 33$ , $a = 77$ and $A = 120^{\circ}$ . Find $c$ exactly. Find the exact angle from three sides In a triangle, $c = 11$ , $a = 60$ and $b = 61$ . Find $B$ . In a triangle, $c = 32$ , $a = 45$ and $b = 67$ . Find $B$ . In a triangle, $a = 35$ , $b = 48$ and $c = 43$ . Find $C$ . Find a side or an angle with the law of cosines, to the nearest tenth In a triangle, $c = 5$ , $a = 47$ and $b = 48$ . Find $B$ to the nearest tenth of a degree. In a triangle, $b = 35$ , $c = 17$ and $a = 31$ . Find $A$ to the nearest tenth of a degree. In a triangle, $b = 46$ , $c = 44$ and $a = 84$ . Find $A$ to the nearest tenth of a degree. Solve bearing problems A ship leaves port and sails 10 mi on a course of S $15^{\circ}$ E, then turns $110^{\circ}$ to the left and goes 60 mi. To the nearest tenth, give the ship's distance from port and the ship's bearing from port. Give the bearing as a three-figure bearing, measured clockwise from north (like ${045.0}^{\circ}$ ). A boat leaves the harbor and travels 45 mi on a course of S $75^{\circ}$ W, then turns $75^{\circ}$ to the right and goes 25 mi. How far is the boat from the harbor, and on what bearing? Round both to the nearest tenth. Give the bearing as a three-figure bearing, measured clockwise from north (like ${045.0}^{\circ}$ ). From the airport, the plane flies 115 mi on a course of S $35^{\circ}$ W. The plane then turns $115^{\circ}$ to the right and goes 60 mi more. How far from the airport does the plane end up, and in what direction? Give the distance and the bearing to the nearest tenth. Give the bearing as a three-figure bearing, measured clockwise from north (like ${045.0}^{\circ}$ ). Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Laws of Sines and Cosines

Date Period

Find each measure asked for.

Use the law of sines
In $△ A B C$ , $∠ B = 120^{\circ}$ , the side opposite it is $36$ , and the side opposite $∠ C$ is $12 \sqrt{6}$ . Find $∠ A$ .
In $△ A B C$ , $a = 35$ , $b = 21$ and $m ∠ A = 42^{\circ}$ . What is $m ∠ B$ , to the nearest tenth of a degree?
In a triangle, $B = 45^{\circ}$ , $b = 28$ and $a = 14 \sqrt{2}$ . Find $C$ .
$△ A B C$ has $m ∠ B = 27^{\circ}$ , $a = 16$ and $b = 21$ . Find $m ∠ A$ to the nearest tenth of a degree.
A triangle has a $60^{\circ}$ angle facing a side of $42$ and a second side of $14 \sqrt{6}$ . Find the angle between those two sides.
In $△ A B C$ , $m ∠ A = 75^{\circ}$ , $a = 15$ and $c = 5$ . Use the law of sines to find the measure of $∠ C$ , to the nearest tenth of a degree.
In $△ A B C$ , $∠ A = 120^{\circ}$ , the side opposite it is $48$ , and the side opposite $∠ B$ is $16 \sqrt{3}$ . Find $∠ C$ .

Use the law of cosines to find the third side exactly
In a triangle, $b = 55$ , $c = 50$ and $A = 90^{\circ}$ . Find $a$ exactly.
In a triangle, $a = 63$ , $b = 65$ and $C = 120^{\circ}$ . Find $c$ exactly.
In a triangle, $b = 68$ , $c = 36$ and $A = 90^{\circ}$ . Find $a$ exactly.
In a triangle, $b = 33$ , $a = 77$ and $A = 120^{\circ}$ . Find $c$ exactly.

Find the exact angle from three sides
In a triangle, $c = 11$ , $a = 60$ and $b = 61$ . Find $B$ .
In a triangle, $c = 32$ , $a = 45$ and $b = 67$ . Find $B$ .
In a triangle, $a = 35$ , $b = 48$ and $c = 43$ . Find $C$ .

Find a side or an angle with the law of cosines, to the nearest tenth
In a triangle, $c = 5$ , $a = 47$ and $b = 48$ . Find $B$ to the nearest tenth of a degree.
In a triangle, $b = 35$ , $c = 17$ and $a = 31$ . Find $A$ to the nearest tenth of a degree.
In a triangle, $b = 46$ , $c = 44$ and $a = 84$ . Find $A$ to the nearest tenth of a degree.

Solve bearing problems
A ship leaves port and sails 10 mi on a course of S $15^{\circ}$ E, then turns $110^{\circ}$ to the left and goes 60 mi. To the nearest tenth, give the ship's distance from port and the ship's bearing from port. Give the bearing as a three-figure bearing, measured clockwise from north (like ${045.0}^{\circ}$ ).
A boat leaves the harbor and travels 45 mi on a course of S $75^{\circ}$ W, then turns $75^{\circ}$ to the right and goes 25 mi. How far is the boat from the harbor, and on what bearing? Round both to the nearest tenth. Give the bearing as a three-figure bearing, measured clockwise from north (like ${045.0}^{\circ}$ ).
From the airport, the plane flies 115 mi on a course of S $35^{\circ}$ W. The plane then turns $115^{\circ}$ to the right and goes 60 mi more. How far from the airport does the plane end up, and in what direction? Give the distance and the bearing to the nearest tenth. Give the bearing as a three-figure bearing, measured clockwise from north (like ${045.0}^{\circ}$ ).



Sigma Prep · sigmaprep.io/worksheets Name The Laws of Sines and CosinesAnswer keyVersion 2 Date Period Find each measure asked for. Use the law of sines In $△ A B C$ , $∠ B = 120^{\circ}$ , the side opposite it is $36$ , and the side opposite $∠ C$ is $12 \sqrt{6}$ . Find $∠ A$ . $15^{\circ}$ In $△ A B C$ , $a = 35$ , $b = 21$ and $m ∠ A = 42^{\circ}$ . What is $m ∠ B$ , to the nearest tenth of a degree? ${23.7}^{\circ}$ In a triangle, $B = 45^{\circ}$ , $b = 28$ and $a = 14 \sqrt{2}$ . Find $C$ . $105^{\circ}$ $△ A B C$ has $m ∠ B = 27^{\circ}$ , $a = 16$ and $b = 21$ . Find $m ∠ A$ to the nearest tenth of a degree. ${20.2}^{\circ}$ A triangle has a $60^{\circ}$ angle facing a side of $42$ and a second side of $14 \sqrt{6}$ . Find the angle between those two sides. $75^{\circ}$ In $△ A B C$ , $m ∠ A = 75^{\circ}$ , $a = 15$ and $c = 5$ . Use the law of sines to find the measure of $∠ C$ , to the nearest tenth of a degree. ${18.8}^{\circ}$ In $△ A B C$ , $∠ A = 120^{\circ}$ , the side opposite it is $48$ , and the side opposite $∠ B$ is $16 \sqrt{3}$ . Find $∠ C$ . $30^{\circ}$ Use the law of cosines to find the third side exactly In a triangle, $b = 55$ , $c = 50$ and $A = 90^{\circ}$ . Find $a$ exactly. $5 \sqrt{221}$ In a triangle, $a = 63$ , $b = 65$ and $C = 120^{\circ}$ . Find $c$ exactly. $\sqrt{12289}$ In a triangle, $b = 68$ , $c = 36$ and $A = 90^{\circ}$ . Find $a$ exactly. $4 \sqrt{370}$ In a triangle, $b = 33$ , $a = 77$ and $A = 120^{\circ}$ . Find $c$ exactly. $55$ Find the exact angle from three sides In a triangle, $c = 11$ , $a = 60$ and $b = 61$ . Find $B$ . $90^{\circ}$ In a triangle, $c = 32$ , $a = 45$ and $b = 67$ . Find $B$ . $120^{\circ}$ In a triangle, $a = 35$ , $b = 48$ and $c = 43$ . Find $C$ . $60^{\circ}$ Find a side or an angle with the law of cosines, to the nearest tenth In a triangle, $c = 5$ , $a = 47$ and $b = 48$ . Find $B$ to the nearest tenth of a degree. ${98.6}^{\circ}$ In a triangle, $b = 35$ , $c = 17$ and $a = 31$ . Find $A$ to the nearest tenth of a degree. ${62.3}^{\circ}$ In a triangle, $b = 46$ , $c = 44$ and $a = 84$ . Find $A$ to the nearest tenth of a degree. ${137.9}^{\circ}$ Solve bearing problems A ship leaves port and sails 10 mi on a course of S $15^{\circ}$ E, then turns $110^{\circ}$ to the left and goes 60 mi. To the nearest tenth, give the ship's distance from port and the ship's bearing from port. Give the bearing as a three-figure bearing, measured clockwise from north (like ${045.0}^{\circ}$ ). $57.4$ mi on a bearing of ${064.4}^{\circ}$ A boat leaves the harbor and travels 45 mi on a course of S $75^{\circ}$ W, then turns $75^{\circ}$ to the right and goes 25 mi. How far is the boat from the harbor, and on what bearing? Round both to the nearest tenth. Give the bearing as a three-figure bearing, measured clockwise from north (like ${045.0}^{\circ}$ ). $56.9$ mi on a bearing of ${280.1}^{\circ}$ From the airport, the plane flies 115 mi on a course of S $35^{\circ}$ W. The plane then turns $115^{\circ}$ to the right and goes 60 mi more. How far from the airport does the plane end up, and in what direction? Give the distance and the bearing to the nearest tenth. Give the bearing as a three-figure bearing, measured clockwise from north (like ${045.0}^{\circ}$ ). $104.8$ mi on a bearing of ${246.2}^{\circ}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

The Laws of Sines and CosinesAnswer keyVersion 2

Date Period

Find each measure asked for.

Use the law of sines
In $△ A B C$ , $∠ B = 120^{\circ}$ , the side opposite it is $36$ , and the side opposite $∠ C$ is $12 \sqrt{6}$ . Find $∠ A$ . $15^{\circ}$
In $△ A B C$ , $a = 35$ , $b = 21$ and $m ∠ A = 42^{\circ}$ . What is $m ∠ B$ , to the nearest tenth of a degree? ${23.7}^{\circ}$
In a triangle, $B = 45^{\circ}$ , $b = 28$ and $a = 14 \sqrt{2}$ . Find $C$ . $105^{\circ}$
$△ A B C$ has $m ∠ B = 27^{\circ}$ , $a = 16$ and $b = 21$ . Find $m ∠ A$ to the nearest tenth of a degree. ${20.2}^{\circ}$
A triangle has a $60^{\circ}$ angle facing a side of $42$ and a second side of $14 \sqrt{6}$ . Find the angle between those two sides. $75^{\circ}$
In $△ A B C$ , $m ∠ A = 75^{\circ}$ , $a = 15$ and $c = 5$ . Use the law of sines to find the measure of $∠ C$ , to the nearest tenth of a degree. ${18.8}^{\circ}$
In $△ A B C$ , $∠ A = 120^{\circ}$ , the side opposite it is $48$ , and the side opposite $∠ B$ is $16 \sqrt{3}$ . Find $∠ C$ . $30^{\circ}$

Use the law of cosines to find the third side exactly
In a triangle, $b = 55$ , $c = 50$ and $A = 90^{\circ}$ . Find $a$ exactly. $5 \sqrt{221}$
In a triangle, $a = 63$ , $b = 65$ and $C = 120^{\circ}$ . Find $c$ exactly. $\sqrt{12289}$
In a triangle, $b = 68$ , $c = 36$ and $A = 90^{\circ}$ . Find $a$ exactly. $4 \sqrt{370}$
In a triangle, $b = 33$ , $a = 77$ and $A = 120^{\circ}$ . Find $c$ exactly. $55$

Find the exact angle from three sides
In a triangle, $c = 11$ , $a = 60$ and $b = 61$ . Find $B$ . $90^{\circ}$
In a triangle, $c = 32$ , $a = 45$ and $b = 67$ . Find $B$ . $120^{\circ}$
In a triangle, $a = 35$ , $b = 48$ and $c = 43$ . Find $C$ . $60^{\circ}$

Find a side or an angle with the law of cosines, to the nearest tenth
In a triangle, $c = 5$ , $a = 47$ and $b = 48$ . Find $B$ to the nearest tenth of a degree. ${98.6}^{\circ}$
In a triangle, $b = 35$ , $c = 17$ and $a = 31$ . Find $A$ to the nearest tenth of a degree. ${62.3}^{\circ}$
In a triangle, $b = 46$ , $c = 44$ and $a = 84$ . Find $A$ to the nearest tenth of a degree. ${137.9}^{\circ}$

Solve bearing problems
A ship leaves port and sails 10 mi on a course of S $15^{\circ}$ E, then turns $110^{\circ}$ to the left and goes 60 mi. To the nearest tenth, give the ship's distance from port and the ship's bearing from port. Give the bearing as a three-figure bearing, measured clockwise from north (like ${045.0}^{\circ}$ ). $57.4$ mi on a bearing of ${064.4}^{\circ}$
A boat leaves the harbor and travels 45 mi on a course of S $75^{\circ}$ W, then turns $75^{\circ}$ to the right and goes 25 mi. How far is the boat from the harbor, and on what bearing? Round both to the nearest tenth. Give the bearing as a three-figure bearing, measured clockwise from north (like ${045.0}^{\circ}$ ). $56.9$ mi on a bearing of ${280.1}^{\circ}$
From the airport, the plane flies 115 mi on a course of S $35^{\circ}$ W. The plane then turns $115^{\circ}$ to the right and goes 60 mi more. How far from the airport does the plane end up, and in what direction? Give the distance and the bearing to the nearest tenth. Give the bearing as a three-figure bearing, measured clockwise from north (like ${045.0}^{\circ}$ ). $104.8$ mi on a bearing of ${246.2}^{\circ}$

How to do these

In a triangle, $b = 7$ , $c = 8$ and $a = 13$ . Find $A$

Law of cosines for A: 13² = 7² + 8² - 2(7)(8) cos A.
169 = 113 - 112 cos A, so cos A = -56/112 = -1/2.
The angle with cosine -1/2 is 120°.

The answer is $120^{\circ}$ .

Where students go wrong

Dropping the sign of the cosine. When a² is larger than b² + c², cos A comes out negative and A is obtuse; losing the minus sign turns 120° into 60°.

Also called non-right triangles or 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. → $2 \sqrt{6}$
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 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 law of cosines, any angle, rounded: In a triangle, $b = 13$ , $c = 8$ and $A = 146^{\circ}$ . Find $a$ to the nearest tenth. → $20.1$
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. → $28$
Area with any angle, and Heron's formula: Two sides of a triangle are $9$ and $10$ , and the angle between them is $20^{\circ}$ . Find the area to the nearest tenth. → $15.4$
Bearings: A hiker leaves camp and walks 9 mi on a bearing of $030^{\circ}$ , then 11 mi on a bearing of $270^{\circ}$ . What are the hiker's distance and bearing from camp now? Round both to the nearest tenth. → $10.1$ mi on a bearing of ${320.2}^{\circ}$

Questions about these worksheets

Yes. Every law of sines and 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 sines and 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 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 10: the law of sines, a missing side, from two sides and the angle between, a missing angle, from three sides, the law of sines, any angle, rounded, the law of cosines, any angle, rounded, the ambiguous case: how many triangles?, the ambiguous case: every possible angle, area from two sides and the angle between, area with any angle, and heron's formula and bearings. Tick as many as you want and set how many of each, or let it spread them evenly.

Choosing the wrong rule. If you have a side and the angle opposite it, use sines; if you have two sides and the angle between them, only cosines will work. The other slip is forgetting that the correction term is subtracted, not added.

Law of sines and cosines worksheet for Precalculus

Select your difficulty

Use the law of sines

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

Solve bearing problems

Use the law of sines

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

Solve bearing problems

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these law of sines and cosines worksheets free?+

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