Law of Cosines

Trigonometry — Solving Any Triangle

In this lesson:

• Find missing sides and angles with the Law of Cosines
• Decide when to use the Law of Cosines or the Law of Sines
• Connect the formula to the Pythagorean theorem

What You Will Learn Today

1. Solve SAS and SSS triangles with the Law of Cosines; finish with the Law of Sines or angle sum
2. Find a side from SAS
3. Find an angle from SSS
4. Explain the Pythagorean theorem as the case
5. Choose between Cosines and Sines

Why Pythagoras and Sines Fall Short

needs a right angle. The Law of Sines needs a side and its opposite angle.

• SAS: two sides and the included angle, find the third side
• SSS: three sides, find any angle
• Mode check: in degree mode, in radian mode

The Law of Cosines Relates All Parts

The angle in the cosine is opposite the side being solved for.

SAS Example: Acute Included Angle

, , :

SAS Example: Obtuse Included Angle

: negative for angles between 90° and 180°.

Right angle: .

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.

The Law of Cosines works for all triangles. Right triangles are the special case you already know how to handle.

The Correction Term Shrinks or Grows

The term corrects the Pythagorean value. Sides and :

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

, , :

is the cosine, not the angle.

SSS Example: Remaining Angles and Rounding

, so

, so

Sum: , rounding only.

Rounding early to gives and a sum of .

SSS Obtuse Example: Largest Angle First

Sides , , ; is opposite the .

• Negative cosine means an obtuse angle
• The other two angles must be acute

Quick Check: Find the Largest Angle

In triangle XYZ: , , .

Which angle is largest? Find it.

Hint: the largest angle is opposite the longest side.

Choosing the Right Law: Decision Framework

A side with its opposite angle starts the Law of Sines; without one, the Law of Cosines starts. Given , , is SSA, not SAS.

Completing the Triangle With Both Laws

, , , and . Find and .

is opposite the shorter given side, so it is acute.

ACT Practice: Choose the Law, Then Solve

Problem 1: , , . SAS, so and .

Problem 2: , , . SSS, largest angle :

Key Takeaways and Common Mistakes

• works for any triangle
• SAS: find a side. SSS: find an angle with

Watch out:

• Write the minus sign; obtuse cosines are negative
• The cosine angle is opposite the target side
• Square root last; degree mode

Coming Up Next in Trigonometry

Up next: Degree-radian conversion

• Radians as another unit for angles
• Converting between degrees and radians
• Why calculator mode matters