Algebra 2 / Precalculus · Step-by-step guide

Law of Cosines

Use the Law of Cosines for SAS and SSS triangles and connect it to the Pythagorean theorem.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

The Law of Cosines is c2=a2+b2−2abcos⁡C\displaystyle c^2=a^2+b^2-2ab\cos C, where side c is opposite angle C. Use it for SAS to find a side or SSS to find an angle.

c2=a2+b2−2abcos⁡C\displaystyle c^2=a^2+b^2-2ab\cos C
01

How it works

Build the method from meaning before memorizing the moves.

These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.

1

Match the opposite pair

The isolated side and angle in the cosine term must be opposite each other.

2

Use SAS or SSS

SAS gives a direct side calculation; SSS requires isolating cosine before inverse cosine.

3

See the right-triangle connection

When C = 90°, cos C = 0 and the formula becomes the Pythagorean theorem.

02

See the structure

A picture makes the relationships easier to remember.

baseheight
The labeled lengths and right-angle marker connect the diagram to the triangle formulas in the guide.
03

Worked examples

Follow the reason for each line, then try to reproduce it without looking.

Example 1SAS

a=5,b=8,C=60∘\displaystyle a=5,b=8,C=60^\circ. Find c.

  1. c2=25+64−80cos⁡60\displaystyle c^2=25+64-80\cos60.
  2. Since cos⁡60=1/2\displaystyle \cos60=1/2, c2=49\displaystyle c^2=49.
  3. Take the positive square root.
Answerc=7\displaystyle c=7
Example 2SSS angle

Sides 6, 9, and 11; find the angle opposite 11.

  1. 112=62+92−2(6)(9)cos⁡C\displaystyle 11^2=6^2+9^2-2(6)(9)\cos C.
  2. Isolate cos⁡C=−4/108\displaystyle \cos C=-4/108.
  3. Use inverse cosine.
AnswerC≈92.1∘\displaystyle C\approx92.1^\circ
Example 3Included angle

Two known sides meet at angle A.

  1. The unknown opposite side must be a.
  2. Use a2=b2+c2−2bccos⁡A\displaystyle a^2=b^2+c^2-2bc\cos A.
  3. Do not pair A with b or c.
AnswerMatch A with side a
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Pairing the angle with a nonopposite side.

  • Forgetting the negative cosine term.

  • Using the Law of Sines without a known opposite pair.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Law of Sines and Cosines

Practice SAS, SSS, and method selection.

Preview the worksheet

A strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.

06

Practice problems with worked solutions

Solve each problem on paper, use the hint only if needed, then compare every step.

1

Find c when a = 5, b = 7, and C = 60°.

C = 60°a = 5b = 7c = ?
Show hint

Use the form solving for c.

Show worked solution
  1. c2=52+72−2(5)(7)cos⁡60°\displaystyle c^2=5^2+7^2-2(5)(7)\cos60°.
  2. c2=25+49−35=39\displaystyle c^2=25+49-35=39.

Answer: c=39\displaystyle c=\sqrt{39}

2

Find the angle C opposite side 8 when a = 5 and b = 6.

C = ?a = 5b = 6c = 8
Show hint

Solve the formula for cosine C.

Show worked solution
  1. cos⁡C=(52+62−82)/(2(5)(6))\displaystyle \cos C=(5^2+6^2-8^2)/(2(5)(6)).
  2. cos⁡C=−3/60=−0.05\displaystyle \cos C=-3/60=-0.05.
  3. C≈92.9°\displaystyle C\approx92.9°.

Answer: Approximately 92.9°.

3

Find x when two sides are 9 and 12 with included angle 90°.

C = 90°912x
Show hint

Cosine 90° is zero.

Show worked solution
  1. x2=92+122−2(9)(12)(0)\displaystyle x^2=9^2+12^2-2(9)(12)(0).
  2. x2=225\displaystyle x^2=225.

Answer: x=15\displaystyle x=15

4

Classify the angle opposite side 10 in a triangle with other sides 6 and 8.

C = ?6810
Show hint

Compare squares or use cosine.

Show worked solution
  1. 102=62+82=100\displaystyle 10^2=6^2+8^2=100.
  2. Equality means the opposite angle is right.

Answer: 90°; a right angle.

5

Find the largest angle of a triangle with sides 4, 7, and 9.

C = ?479
Show hint

The largest angle is opposite side 9.

Show worked solution
  1. cos⁡C=(42+72−92)/(2(4)(7))=−16/56\displaystyle \cos C=(4^2+7^2-9^2)/(2(4)(7))=-16/56.
  2. C=cos⁡−1(−2/7)≈106.6°\displaystyle C=\cos^{-1}(-2/7)\approx106.6°.

Answer: Approximately 106.6°.

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

The Law of Cosines is c2=a2+b2−2abcos⁡C\displaystyle c^2=a^2+b^2-2ab\cos C, where side c is opposite angle C. Use it for SAS to find a side or SSS to find an angle.

How do I know whether I understand this topic?

You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.

What should I do if I keep making the same mistake?

Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.

Keep learning

Related guides