Trigonometry · Trigonometry: Similar Triangles, Right Triangles and Acute Triangles

The Cosine Law

3.2 3.3About 75 minutes

Learning goals

I can follow how the cosine law is developed using the Pythagorean theorem and the cosine ratio.
I can explain how the cosine law is related to the Pythagorean theorem.
I can use the cosine law to find a side when I know two sides and the contained angle (SAS).
I can use the cosine law to find an angle when I know all three sides (SSS).

Words to know

cosine law
In any △ABC: a² = b² + c² − 2bc cos A (and similarly for b² and c²).

contained angle
The angle between two given sides.

SAS
Two sides and the contained angle are known.

SSS
All three sides are known.

When the sine law gets stuck

The sine law needs a complete pair: a side and the angle opposite it. Sometimes you do not have one. Two hiking trails leave a parking lot at a 55° angle; you know how long each trail is and want the straight distance between their ends (SAS). Or you know all three sides of a triangular garden and want an angle (SSS). For these you need the cosine law, which is the Pythagorean theorem with a correction term.

Where the cosine law comes from

Start with an acute △ABC and draw the altitude h from C to AB, meeting it at D. Let AD = x, so DB = c − x.

In right △ACD: cos A = x/b, so x = b cos A. Also, by Pythagoras, h² = b² − x².
In right △BCD, by Pythagoras: a² = h² + (c − x)².
Substitute h² and expand: a² = b² − x² + c² − 2cx + x².
The x² terms cancel: a² = b² + c² − 2cx.
Replace x with b cos A:

a² = b² + c² − 2bc cos A

You do not have to memorize this development, but notice the ingredients: the Pythagorean theorem (twice) and the cosine ratio.

The altitude splits side c into x and c − x. Two uses of the Pythagorean theorem and one cosine ratio give the cosine law.

The cosine law (three versions)

a² = b² + c² − 2bc cos A

b² = a² + c² − 2ac cos B

c² = a² + b² − 2ab cos C

Pattern: the side on the left is opposite the angle on the right; the other two sides appear everywhere else.

To find an angle, rearrange: cos A = (b² + c² − a²) ÷ (2bc).

The connection to Pythagoras

If ∠A = 90°, then cos A = cos 90° = 0, and the cosine law becomes a² = b² + c²: the Pythagorean theorem! For an acute angle, cos A is positive, so the term 2bc cos A is subtracted, and the side opposite the angle is shorter than it would be in a right triangle. The cosine law is the Pythagorean theorem plus a correction for angles that are not 90°.

Same two sides (5 and 6), different angles between them. Closing the angle from 90° to 60° shortens the opposite side from about 7.8 to about 5.6. The cosine law measures exactly how much.

Worked example

Example 1: SAS, finding a side

In △ABC, b = 7 cm, c = 9 cm and ∠A = 48°. Find a.

Two sides and the contained angle: cosine law for a.
a² = 7² + 9² − 2(7)(9) cos 48°.
a² = 49 + 81 − 126 cos 48° = 130 − 84.31... = 45.69...
a = √45.69... = 6.759... ≈ 6.8 cm.
Order of operations: multiply 2 × 7 × 9 × cos 48° first, then subtract from 130.

a ≈ 6.8 cm

Worked example

Example 2: SSS, finding an angle

A triangle has sides a = 8 m, b = 10 m and c = 12 m. Find ∠C (the largest angle).

The largest angle is opposite the longest side, c = 12.
cos C = (a² + b² − c²) ÷ (2ab) = (64 + 100 − 144) ÷ 160 = 20 ÷ 160 = 0.125.
∠C = cos⁻¹(0.125) = 82.8...° ≈ 83°.
The largest angle is less than 90°, so this is an acute triangle. ✓

∠C ≈ 83°

Worked example

Example 3: Two park paths

Two straight paths leave a park gate at an angle of 55°. One is 120 m long and the other is 150 m long. How far apart are the ends of the paths?

SAS: two sides (120 m and 150 m) and the contained angle (55°).
d² = 120² + 150² − 2(120)(150) cos 55° = 14 400 + 22 500 − 36 000 cos 55°.
d² = 36 900 − 20 648.7... = 16 251.2...
d ≈ 127.5 m.

The ends are about 127.5 m apart.

Example 3, drawn to scale. Two sides and the angle between them: a job for the cosine law.

Is the triangle acute?

The cosine law can also tell you the type of triangle. Look at the longest side c:

if c² < a² + b², the largest angle is acute (so the whole triangle is acute);
if c² = a² + b², it is a right triangle;
if c² > a² + b², it is obtuse (you will handle these in Grade 11).

Just for fun

The cosine law is the Pythagorean theorem's older, more flexible sibling. Pythagoras only works at exactly 90°. The cosine law shrugs and says, "Any angle? No problem."

Hands-on activity

Door-angle experiment

You need: a door that swings open, a measuring tape, masking tape and a protractor (or a phone level/angle app). Keep fingers away from the hinge side.

Measure the width of the door, w (from the hinge to the outer edge).
Open the door to about 60°. Check the angle between the door and the closed position with the protractor on the floor.
With masking tape, mark on the floor where the outer edge of the door is when closed and where it is now.
Predict the distance between the two tape marks with the cosine law: d² = w² + w² − 2(w)(w) cos(angle).
Measure the distance between the marks. Repeat for about 40° and 80°.

What to look for: for an 80 cm door opened 60°, the triangle is equilateral, so d = 80 cm. At 40°, d² = 2(80²)(1 − cos 40°) gives d ≈ 54.7 cm. Measurements within 1 or 2 cm are good.

Your turn

In △ABC, b = 5, c = 8 and ∠A = 60°. Find a.

In △ABC, a = 10 cm, b = 12 cm and ∠C = 50°. Find c.

A triangle has sides a = 6, b = 7 and c = 8. Find ∠C.

A triangle has sides a = 9, b = 10 and c = 11. Find ∠A.

In △PQR, p = 15 m, q = 13 m and ∠R = 70°. Find r.

Show that the cosine law becomes the Pythagorean theorem when ∠C = 90°.

You know all three sides of a triangle. Which law do you use to find an angle?

More practice: warm-up

Write the cosine law for c².

What is cos 90°?

Evaluate 3² + 4² − 2(3)(4) cos 60°.

Which two situations need the cosine law?

Rearrange the cosine law to give cos A.

More practice: on your own

A triangular garden has two sides of 12 m and 15 m with a 70° angle between them. How much fencing is needed to go all the way around?

Two ships leave port at the same time. One sails at 15 km/h and the other at 20 km/h, on courses 65° apart. How far apart are they after 2 hours?

A triangle has sides 7 cm, 8 cm and 9 cm. Find all three angles.

Lin computed a² = (7² + 9² − 2 × 7 × 9) cos 48° for Example 1 and got a ≈ 1.6. What went wrong?

Is a triangle with sides 5, 6 and 7 acute? Use the cosine law.

Challenge

Use the cosine law to decide whether a triangle with sides 6, 8 and 11 is acute.

A parallelogram has sides 8 cm and 5 cm, with a 60° angle between them. Find the length of its shorter diagonal.

An isosceles triangle has two sides of 10 cm with a 40° angle between them. Find the third side with the cosine law. (Compare with the Lesson 6 challenge about an isosceles triangle with base angles of 70°.)

Check your understanding

Quiz

Which is the cosine law for side a?

In △ABC, b = 5, c = 8 and ∠A = 60°. What is a?

You know all three sides and want an angle. Which tool?

Why does the cosine law become the Pythagorean theorem when the angle is 90°?

A triangle has sides 8, 10 and 12. What is the angle opposite the side of length 12?

To evaluate 130 − 126 cos 48°, what should you do first?

For parents and teachers

Big idea: Expectations 3.2 and 3.3: exploring the development of the cosine law, seeing how it connects to the Pythagorean theorem and the cosine ratio, and using it in acute triangles. Students do not have to reproduce the development.

How to help: Ask "Is there a complete pair (side + opposite angle)?" If yes, the sine law works; if not (SAS or SSS), use the cosine law. Have your teen write the full substitution before pressing calculator keys.

Common difficulties:

Order of operations: multiplying (b² + c² − 2bc) by cos A, or subtracting before multiplying. Compute 2bc cos A as one chunk.
Forgetting the square root at the end when finding a side.
Mixing up the letters: the angle in the formula must be opposite the side on the left.
Calculator in radian mode.