Law of Cosines- Finding Sides and Angles

What the Law of Cosines Actually Is

The Law of Cosines is a formula that connects three sides of a triangle to one of its angles. It's not complicated. It's just an extension of the Pythagorean theorem that works when you don't have a right triangle.

Most students encounter this after mastering SOH CAH TOA and the basic trig ratios. The problem is, those tools only work on right triangles. Real triangles don't always cooperate.

This is where the Law of Cosines steps in. It handles triangles where you know two sides and the angle between them (SAS), or all three sides (SSS). Both situations make SOH CAH TOA useless.

The Formula

For any triangle with sides a, b, and c, where c is the side opposite angle C:

c² = a² + b² - 2ab·cos(C)

The pattern repeats for any side. If you're solving for side a, it's:

a² = b² + c² - 2bc·cos(A)

Notice the structure. You square two sides, subtract twice their product times the cosine of the included angle, and you get the square of the opposite side. That's it.

When to Use This (And When Not To)

You need the Law of Cosines in two specific situations:

• SAS (Side-Angle-Side): You know two sides and the angle between them
• SSS (Side-Side-Side): You know all three sides and need to find an angle

If you have a right triangle, skip this entirely. The Pythagorean theorem is faster and simpler. If you know two angles and one side (AAS or ASA), use the Law of Sines instead.

Law of Cosines vs. Law of Sines

Here's a quick comparison:

Most students waste time trying to force the Law of Sines into SAS or SSS problems. It doesn't work. Use the right tool for the job.

Finding a Side: Step-by-Step

Say you have a triangle where a = 7, b = 10, and the included angle C = 45°. You need to find side c.

Step 1: Plug into the formula

c² = 7² + 10² - 2(7)(10)·cos(45°)

Step 2: Calculate the squares

c² = 49 + 100 - 140·cos(45°)

Step 3: Evaluate the cosine. cos(45°) = 0.7071

c² = 149 - 140(0.7071)

c² = 149 - 98.99

c² = 50.01

Step 4: Take the square root

c = √50.01 ≈ 7.07

That's it. Square, multiply, subtract, square root. The calculator work is where people mess up, not the formula itself.

Finding an Angle: The Rearranged Formula

When you know all three sides and need an angle, rearrange the formula to solve for cosine first:

cos(C) = (a² + b² - c²) / (2ab)

Then use inverse cosine (arccos or cos⁻¹) to find the angle.

Example: Find angle C if a = 8, b = 6, and c = 5.

cos(C) = (8² + 6² - 5²) / (2 × 8 × 6)

cos(C) = (64 + 36 - 25) / 96

cos(C) = 75 / 96

cos(C) = 0.78125

C = cos⁻¹(0.78125) ≈ 38.6°

Check yourself: the largest side is usually opposite the largest angle. Here, side a = 8 is the longest, so angle A should be larger than 38.6°. That tracks.

Common Mistakes That Will Cost You Points

• Using the wrong sign: The formula has a minus sign before 2ab·cos(C). Students forget this and get answers that make no sense.
• Rounding too early: Keep full calculator precision until the final answer. Rounding at each step compounds errors.
• Confusing which side goes where: The angle in the cosine term must be between the two sides you're multiplying. Always verify this before plugging in numbers.
• Forgetting to check your answer: Plug your result back into the original formula. If it doesn't work, you made an error somewhere.

Practical How-To: Solving Any Law of Cosines Problem

1. Identify what you know

Write down the three sides and three angles. Circle what you know. Box what you need to find.

2. Pick the right formula version

If finding a side: use c² = a² + b² - 2ab·cos(C) with C as the included angle.

If finding an angle: use cos(C) = (a² + b² - c²) / (2ab) where c is opposite angle C.

3. Plug in your numbers

Double-check each value before calculating. One wrong number ruins everything.

4. Calculate carefully

Use your calculator's memory function if needed. Watch your parentheses.

5. Verify your answer

Does the found side make sense? Is the angle between 0° and 180°? Does your answer satisfy the original problem?

The Bottom Line

The Law of Cosines is not difficult. The formula is straightforward. The math is basic algebra with one trig function. What trips people up is knowing when to apply it and keeping track of which side goes where.

Master SAS and SSS cases. Practice rearranging the formula for angle problems. Check your answers every time. Do that, and these problems become routine. 🔺