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Picture of Law Of Cosines Formula

The Law of Cosines

Formula and examples of law of cosines

Law of cosines worksheet

(25 question worksheet with answer key)
Law of sines vs cosines
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
Law of Cosines Formula and Picture

When to use law of cosines?

question
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.

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Example 1 , Given : 2 sides and 1 angle


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


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.

Answer


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

Answer


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

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


problem 5) Look at the the three triangles below. For which one(s) can you use the law of cosines to find the length of the unknown side , side a ?
Answer

identify formula use case
identify formula use case
identify formula use case
The law of Cosines and the Pythagorean theorem.
Use the law of cosines to calculate the value of x. This exercise should help you see the connection between the law of cosines and the Pythagorean theorem.
Answer
Challenge problems
Law of Cosines