Law of Cosines

Trigonometry — Solving Any Triangle

In this lesson:

• Apply the Law of Cosines to find missing sides and angles
• Decide when to use Law of Cosines vs. Law of Sines
• Connect the formula to the Pythagorean theorem

What You Will Learn Today

After this lesson, you will be able to:

1. State the Law of Cosines formula
2. Find a missing side given two sides and the included angle
3. Find a missing angle given all three sides
4. Explain why the Pythagorean theorem is a special case

Why the Pythagorean Theorem Falls Short

You know — but only for right triangles.

• What if the triangle has no right angle?
• What if you know two sides and the included angle?
• The Law of Sines needs an opposite side-angle pair

The Law of Cosines Relates All Parts

The angle in the cosine is always opposite the target side.

When to Use the Law of Cosines

SAS — Two sides and the included angle → find the third side

SSS — All three sides known → find any angle

Key question: Do I have an opposite side-angle pair?

• Yes → Law of Sines
• No → Law of Cosines

SAS Example: Finding a Side (Acute Angle)

In triangle ABC: , , . Find side .

Step 1: Write the formula

SAS Acute Example: Complete the Calculation

Keep 4 decimal places in intermediate steps. Take the square root last.

SAS Example: Finding a Side (Obtuse Angle)

In triangle DEF: , , . Find side .

When the angle is obtuse, is negative — the side gets longer.

Quick Check: Find the Missing Side

In triangle PQR: , , .

Find side .

Set up the formula and compute before advancing...

Pythagorean Theorem as a Special Case

When , :

The correction term vanishes — leaving the Pythagorean theorem.

• Acute angle: correction negative, is shorter
• Right angle: correction zero, Pythagorean theorem
• Obtuse angle: correction positive, is longer

Rearranging to Find a Missing Angle

To find angle when all three sides are known, rearrange:

Then apply inverse cosine:

No ambiguous case — inverse cosine returns a unique angle in .

SSS Example: Find Angle C Step by Step

In triangle ABC: , , . Find angle .

SSS Example: Verify Angles Sum to 180

Find angle using :

Angle B:

Check: ✓

Quick Check: Find the Largest Angle

In triangle XYZ: , , .

Which angle is largest? Find it.

Hint: the largest angle is opposite the longest side.

Strategy: Find the Largest Angle First

Why? The largest angle is opposite the longest side.

• If it is obtuse, the other two must be acute
• This avoids ambiguity in remaining calculations
• After two angles, get the third from

Tip: Use Law of Cosines for the first angle, then subtract.

Choosing the Right Law: Decision Framework

Ask: Do I have a complete opposite side-angle pair?

ACT Practice: Identify and Solve These

Problem 1: In , , , . Find .

Problem 2: In , , , . Find the largest angle.

Identify SAS or SSS, choose the formula, then solve.

Combined-Law Problem: Using Both Laws Together

In : , , . Find angle .

Step 1: Law of Cosines for

Step 2: Law of Sines for

ACT Practice Solutions: Check Your Answers

Problem 1: SAS →

Problem 2: SSS, largest angle opposite :

Key Takeaways and Common Mistakes

Remember:

• — any triangle
• SAS → find side; SSS → find angle

Watch out:

• Write the minus sign — obtuse cosines are negative
• angle is opposite the target side
• Take the square root at the end

Coming Up Next in Trigonometry

Up next: Trigonometric applications and word problems

• Distance and navigation problems using both laws
• Indirect measurement with triangulation
• Multi-step ACT problems combining all trig tools