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The Law of Cosines

Formula and examples of law of cosines

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Law of Cosines Formula
The law of cosines is a formula that relates the three sides of a triangle to the cosine of a given angle

When to use law of cosines?

Interactive Demonstration of the Law of Cosines Formula
The interactive demonstration below illustrates the Law of cosines formula in action. Drag around the points in the triangle to observe who the formula works.
Example 1 , Given : 2 sides and 1 angle

$$b^2 = a^2 + c^2 - 2ac\cdot \text{cos}(44) \\ \color{red}{x}^2 = 14^2 + 10^2 -2 \cdot 14 \cdot 10 \text{cos}(44 ^ \circ ) \\ \color{red}{x}^2 = 14^2 + 10^2 -2 \cdot 14 \cdot 10 \text{cos}(44 ^ \circ ) \\ \color{red}{x}^2 = 296 -280 \text{cos}(44 ^ \circ) \\ \color{red}{x}^2 = 94.5848559051777 \\ \color{red}{x} = \sqrt{ 94.5848559051777} \\ \color{red}{x} = 9.725474585087234$$

Example 2. Given : 3 sides

$$a^2 = b^2 + c^2 - 2bc\cdot \text{cos}(\color{red}{A}) \\ 25^2 = 32^2 + 37^2 -2 \cdot 32 \cdot 37 \cdot \text{cos}(\color{red}{A}) \\ 625 =2393 - 2368\cdot \text{cos}(\color{red}{A}) \\ \frac{625-2393}{ - 2368}= cos(\color{red}{A}) \\ 0.7466216216216216 = cos(\color{red}{A}) \\ \color{red}{A} = cos^{-1} (0.7466216216216216 ) \\ \color{red}{A} = 41.70142633732469 ^ \circ$$
Practice Problems

The problems below are relatively straightforward ones that should help you become more comfortable using this formula. If they seem too easy, move down to the second set of problems .
Problem 1) Use the law of cosines formula to calculate the length of side C.

Problem 2) Use the law of cosines formula to calculate the measure of $$\angle x$$

Problem 3) Use the law of cosines formula to calculate the length of side b.

Problem 4) Use the law of cosines formula to calculate X.