Geometry Law of Sines and Law of Cosines Worksheet (PDF)

Geometry math worksheet: Law of Sines and Law of Cosines – Maths PDF

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This law of sines cosines worksheet for Geometry covers law of sines and law of cosines (Common Core G-SRT.D.9, G-SRT.D.10, G-SRT.D.11), with a free PDF to print.

These 20 problems practice the area formula with sine, the Law of Sines and the Law of Cosines (Common Core G-SRT.D.9 to G-SRT.D.11). The first problems are quick one-step checks of the definitions and formulas from the lesson. The middle problems ask for diagrams, equations and careful reasoning in the style of a high school Geometry class. The last six problems are applications with lakes, ships, lookouts, roofs and lots.

Geometry · Common Core State Standards: G-SRT.D.9, G-SRT.D.10, G-SRT.D.11

Law of sines cosines worksheet: exercises 1 to 10

Exercise 1 – Area from SAS

Area = (1/2)(8)(11) sin 30° = 22.

Two sides of a triangle are 8 and 11, and the angle between them is $30^{\circ}$. Find the area.

Exercise 2 – Area of a sail

A triangular sail has two edges of 7.5 meters and 12 meters with a $58^{\circ}$ angle between them. Find the area of the sail to the nearest tenth of a square meter.

Exercise 3 – Law of Sines: find a side

Law of Sines: a / sin A = b / sin B = c / sin C.

In triangle ABC, $A = 35^{\circ}$, $B = 72^{\circ}$ and $a = 15$. Find b to the nearest tenth.

Exercise 4 – Law of Sines: find an angle

In triangle ABC, $a = 18$, $b = 15$ and $A = 50^{\circ}$. Find angle B to the nearest tenth. (Side b is shorter than a, so B is acute.)

Exercise 5 – ASA

In triangle ABC, $A = 52^{\circ}$, $B = 61^{\circ}$ and $c = 20$. Find angle C and side a to the nearest tenth.

Exercise 6 – Law of Cosines: SAS

Two sides and the included angle.

Two sides of a triangle measure 12 and 15, and the angle between them is $48^{\circ}$. Find the third side c to the nearest tenth.

Exercise 7 – Law of Cosines: SSS

Given three sides, use the Law of Cosines to find an angle.

A triangle has sides 5, 7 and 9. Find the angle opposite the side of length 5 to the nearest tenth.

Exercise 8 – Obtuse angle

A triangle has sides 4, 5 and 8. Find the largest angle to the nearest tenth.

Exercise 9 – Right triangle check

Use the Law of Cosines to show that a triangle with sides 7, 24 and 25 is a right triangle.

Exercise 10 – Which law?

Choose the first step (Law of Sines or Law of Cosines) for each: (a) two angles and a side, (b) two sides and the included angle, (c) three sides, (d) two sides and an angle opposite one of them.

Exercises 11 and up

Exercise 11 – Solve a triangle (SAS)

In triangle ABC, $a = 10$, $b = 12$ and $C = 40^{\circ}$. Find c, then angle A, then angle B. Round to the nearest tenth.

Exercise 12 – Angle opposite the shorter side

In triangle ABC, $A = 58^{\circ}$, $a = 22$ and $b = 18$. Side b is shorter than side a, so angle B is acute. Find the measure of angle B to the nearest tenth of a degree.

Exercise 13 – Equilateral triangle area

Find the area of an equilateral triangle with side 10 using $\frac{1}{2}ab \text{sin}\,C$. Give an exact answer and a decimal.

Exercise 14 – Angle from area

A triangle has area 30 and two sides of 10 and 12 with an acute included angle. Find that angle.

Exercise 15 – Across a lake

Surveyors at C measure to A and B.

Ethan and Mei stand at A and B on opposite shores of a lake. From a point C, they measure CA = 420 feet, CB = 380 feet and angle $C = 48^{\circ}$. Find the distance AB across the lake to the nearest foot.

Exercise 16 – Ship's return

A ship leaves port and sails 40 miles. It turns so that the angle between its path out and its new path is $110^{\circ}$, and then sails 55 miles. How far is the ship from port, to the nearest mile?

Exercise 17 – Fire lookouts

Two lookouts are 800 feet apart. They sight the same fire at angles of $62^{\circ}$ and $71^{\circ}$ from the line joining them. Find the distance from the lookout with the $62^{\circ}$ angle to the fire to the nearest foot.

Exercise 18 – Hikers apart

Two hikers leave the same camp on straight trails. One walks 5.4 km and the other walks 3.8 km, and the angle between the trails is $112^{\circ}$. How far apart are they at the end, to the nearest tenth of a kilometer?

Exercise 19 – Garden lot cost

A triangular lot has two sides of 120 feet and 150 feet meeting at a $35^{\circ}$ angle. Sod costs $0.45 per square foot. Find the area and the cost of covering the lot with sod to the nearest dollar.

Exercise 20 – Order the sides

A triangle has angles of $35^{\circ}$, $65^{\circ}$ and $80^{\circ}$. The side opposite the $35^{\circ}$ angle is 6. Which side is longest? Find the other two sides to the nearest tenth.

Key facts behind this law of sines cosines worksheet

Key facts

Area = (1/2) a b sin C.
Law of Sines: a/sin A = b/sin B = c/sin C.
Law of Cosines: c squared = a squared + b squared – 2ab cos C.
AAS or ASA: use sines. SAS or SSS: use cosines.
When the included angle is 90 degrees the Law of Cosines becomes the Pythagorean theorem.

How to use this law of sines cosines worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the law of sines cosines worksheet as a PDF with the button below and keep it with your notes.

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