Law of Cosines: Step-by-Step Guide
The Law of Cosines is like the Pythagorean theorem, but it works for any triangle: c^2 = a^2 + b^2 - 2ab cos(C). You use it when you know two sides and the angle between them (SAS) or all three sides (SSS), because the Law of Sines does not work directly in those cases.
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Figure out what you know. Use the Law of Cosines for SAS (to find the third side) or SSS (to find an angle).\[\text{SAS or SSS}\]
- Write the formula with the right pairing. The angle in the formula is across from the side on the left side of the equation.\[c^2 = a^2 + b^2 - 2ab\cos C\]
- Put the numbers in and simplify. Substitute the known values and work it out.\[c = \sqrt{a^2 + b^2 - 2ab\cos C}\]
- If you want an angle, rearrange. For SSS, solve the formula for cos(C), then use arccos.\[\cos C = \frac{a^2 + b^2 - c^2}{2ab}\]
- Check. The sides should follow a + b > c, and the angles should add up to 180 degrees.\[a + b > c,\;\;A + B + C = 180^\circ\]
Worked example
In triangle ABC, a = 7, b = 9, and C = 60 deg. Find side c.
Step 1: This is SAS. We know two sides and the angle between them.
Step 2: Use c^2 = a^2 + b^2 - 2ab cos(C).
Step 3: c^2 = 49 + 81 - 2 x 7 x 9 x cos(60 degrees) = 130 - 126 x (1/2) = 130 - 63 = 67.
Step 4: c = the square root of 67, which is about 8.185.
Step 5: Check the triangle inequality: 7 + 9 = 16, which is bigger than 8.185. It works.
The answer is c = the square root of 67, which is about 8.19.
Tips and common mistakes
- If the angle is 90 degrees, cos(90) = 0 and the formula becomes the Pythagorean theorem.
- If cos(C) turns out negative, the angle C is obtuse (bigger than 90 degrees).
- For SSS, after you find one angle, you can use the Law of Sines to find the others.
- Keep your calculator in the same mode (degrees or radians) as the problem.
- Check with the triangle inequality: each side must be shorter than the other two added together.
- It is easy to make mistakes with the squaring and combining. Go slowly.
- For SSS, find the biggest angle first. This avoids the problem of ambiguous answers.
- The formula can be written three ways. Match each angle with the side across from it.