C. Trigonometric Functions · Trigonometry of Acute Triangles

Using the Cosine Law to Find Angles and Solve Triangles

C1.4About 70 minutes

Learning goals

I can rearrange the cosine law to find an angle when I know all three sides (SSS).
I can solve a triangle completely, starting with the cosine law and finishing with the sine law or the angle sum.
I can predict which angle is largest before I calculate.
I can describe when the cosine law is the right tool: SAS or SSS.

Words to know

SSS
All three sides are known.

rearranged cosine law
cos A = (b² + c² − a²) ÷ (2bc), used to find an angle from three sides.

included (contained) angle
The angle formed by two given sides.

Three sides, no angles

A carpenter builds a triangular brace from three boards of known length. A hiker knows the road distances between three towns. In both cases all three sides are known and no angles. The sine law needs an angle to start, so it is stuck again. The cosine law can find an angle, once you rearrange it.

Rearranging the cosine law to make the angle the subject. The same steps work for cos B and cos C.

The cosine law for angles

cos A = (b² + c² − a²) ÷ (2bc)

cos B = (a² + c² − b²) ÷ (2ac)

cos C = (a² + b² − c²) ÷ (2ab)

Pattern: the side across from the angle is the one being subtracted. The two sides in the denominator are the sides that form the angle. Use brackets on your calculator around the top and around the bottom.

Worked example

Example 1: Find the largest angle

A triangular brace has sides 6 m, 7 m and 8 m. Find its largest angle.

The largest angle is across from the longest side (8 m). Call it ∠C, with c = 8, a = 6, b = 7.
cos C = (6² + 7² − 8²) ÷ (2 × 6 × 7) = (36 + 49 − 64) ÷ 84 = 21 ÷ 84 = 0.25
∠C = cos⁻¹(0.25) ≈ 75.5°
The largest angle is less than 90°, so the triangle is acute. ✓

The largest angle is about 76°.

Example 1, to scale. Predict first: the biggest angle faces the longest side.

Worked example

Example 2: Solve a triangle from three sides

Solve △ABC with a = 12.4 cm, b = 10.8 cm and c = 13.6 cm.

cos A = (10.8² + 13.6² − 12.4²) ÷ (2 × 10.8 × 13.6) = (116.64 + 184.96 − 153.76) ÷ 293.76 ≈ 0.5033, so ∠A ≈ 59.8°.
cos B = (12.4² + 13.6² − 10.8²) ÷ (2 × 12.4 × 13.6) = (153.76 + 184.96 − 116.64) ÷ 337.28 ≈ 0.6584, so ∠B ≈ 48.8°.
∠C = 180° − 59.8° − 48.8° ≈ 71.4°.
Check: the longest side c is across from the largest angle ∠C. ✓ (You could also find ∠B with the sine law once ∠A is known.)

∠A ≈ 60°, ∠B ≈ 49°, ∠C ≈ 71°

Worked example

Example 3: Three towns

By straight-line distance, Queensville is 37 km from Riverton and 42 km from Pinecrest, and Riverton is 50 km from Pinecrest. (These are made-up towns.) At Queensville, what is the angle between the lines to the other two towns?

The angle at Queensville is across from the Riverton–Pinecrest side (50 km).
cos Q = (37² + 42² − 50²) ÷ (2 × 37 × 42) = (1369 + 1764 − 2500) ÷ 3108 = 633 ÷ 3108
cos Q ≈ 0.2037, so ∠Q ≈ 78.2°.

The angle at Queensville is about 78°.

Example 3, to scale. The angle at Queensville is across from the 50 km side, so 50² is the term that gets subtracted.

Worked example

Example 4: SAS, then finish the triangle

Solve △ABC with b = 9.0 m, c = 12.0 m and ∠A = 58°.

Cosine law for the side: a² = 81 + 144 − 216 cos 58° ≈ 110.54, so a ≈ 10.51 m.
Now there is a complete pair (a, ∠A). Use the sine law for ∠B: sin B = 9.0 sin 58° ÷ 10.51 ≈ 0.7260, so ∠B ≈ 46.5°.
∠C = 180° − 58° − 46.5° ≈ 75.5°.
Tip: finding the angle across from the shorter known side (b = 9) with the sine law is the safe habit. In this unit all triangles are acute, so either choice works.

a ≈ 10.5 m, ∠B ≈ 47°, ∠C ≈ 75°

Just for fun

A funny-but-true fact: if the three "sides" you are given are 3, 4 and 10, the cosine law gives cos θ = (9 + 16 − 100) ÷ 24, which is less than −1. Your calculator will refuse. Triangles have rules: the two shorter sides must add up to more than the longest one.

Hands-on activity

Find angles from three measured lengths

You need: three pieces of string, uncooked spaghetti or straws cut to different lengths (for example 15 cm, 18 cm and 20 cm), a ruler, a protractor, tape and a calculator.

Measure the three pieces and tape them into a triangle on paper.
Use the cosine law to calculate all three angles. Check that they add to 180°.
Measure the angles with a protractor and compare.
Try a triangle where the longest piece is almost as long as the other two together. What happens to the largest angle?

What to look for: calculated and measured angles within 2° or 3°. When the pieces barely make a triangle, one angle gets close to 180° (that triangle is no longer acute) and the other two get very small.

Your turn

Find ∠A in △ABC if a = 7, b = 8 and c = 9.

A triangle has sides 10 cm, 11 cm and 12 cm. Find its largest angle.

Solve △PQR: p = 5.2 m, q = 6.1 m, r = 4.8 m.

Solve △ABC: a = 20 cm, c = 24 cm, ∠B = 66°.

Without calculating, which angle of a triangle with sides 4, 5 and 6 is the smallest?

A triangular garden bed has sides 3.0 m, 3.5 m and 4.0 m. Find the angle between the 3.0 m and 3.5 m sides.

More practice: warm-up

Write cos B in terms of the sides of △ABC.

Evaluate (25 + 36 − 16) ÷ 60.

Find cos⁻¹(0.5).

Is SSS a case for the sine law or the cosine law?

Which side is subtracted when finding ∠C?

More practice: on your own

Three trails form a triangle with lengths 2.4 km, 3.1 km and 2.8 km. Find the angle between the 2.4 km and 2.8 km trails.

Liam calculated cos A = 10.8² + 13.6² − 12.4² ÷ 2 × 10.8 × 13.6 without brackets and got a huge number. Explain.

Solve △XYZ: x = 15, y = 17, z = 13.

A triangle has sides 5, 12 and 13. Use the cosine law to find the largest angle. What do you notice?

An equilateral-looking triangle has sides 9.8 cm, 10.0 cm and 10.3 cm. Find its largest angle to one decimal place.

Challenge

Use the cosine law to explain why a triangle with sides a, b, c is acute exactly when the square of the longest side is less than the sum of the squares of the other two.

Decide whether triangles with these sides are acute, right or obtuse: (a) 7, 8, 10 (b) 8, 15, 17 (c) 6, 7, 10.

In an acute triangle with sides 8, 9 and x (where x is the longest side), what values can x have?

Check your understanding

Quiz

Which formula finds ∠A from three sides?

A triangle has sides 6, 7 and 8. Its largest angle is about:

In a triangle with sides 4, 5 and 6, the largest angle is across from:

You know a, b and c. Which tool should you use FIRST?

After using the cosine law to find side a in an SAS problem, what can you use to find ∠B?

A calculator gives an error for cos⁻¹ of your fraction. What is the most likely reason?

For parents and teachers

Big idea: Expectation C1.4: use the cosine law (and the sine law) to calculate sides and angles in acute triangles. This lesson adds SSS (angles from three sides) and full solutions that combine both laws.

How to help: Ask your teen to predict the largest angle before calculating, and to check that the three angles add to 180°.

Common difficulties:

Subtracting the wrong side's square (it must be the side across from the angle).
Missing brackets around the numerator or denominator on the calculator.
Forgetting cos⁻¹ at the end and reporting a cosine value as the angle.