How to Use the Cosine Rule

Step-by-Step Guide

1. 1

   Step 1: Assess what values you know.

   To find the missing side length of a triangle, you need to know the lengths of the other two sides, as well as the size of the angle between them.For example, you might have triangle XYZ.
   
   Side YX is 5 cm long.
   
   Side YZ is 9 cm long.
   
   Angle Y is 89 degrees.
   
   How long is side XZ?
2. 2

   Step 2: Set up the formula for the Cosine Rule.

   This is also called the law of cosines.
   
   The formula is c2=a2+b2−2abcosC{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos {C}}.
   
   In this formula, c{\displaystyle c} equals the missing side length, and cosC{\displaystyle \cos {C}} equals the cosine of the angle opposite the missing side length.
   
   The variables a{\displaystyle a} and b{\displaystyle b} are the lengths of the two known sides., The variables a{\displaystyle a} and b{\displaystyle b} are the two known side lengths.
   
   The variable C{\displaystyle C} is the known angle, which should be the angle between a{\displaystyle a} and b{\displaystyle b}.For example, since the length of side XZ is missing, this side length will stand for c{\displaystyle c} in the formula.
   
   Since sides YX and YZ are known, these two side lengths will be a{\displaystyle a} and b{\displaystyle b}.
   
   It doesn’t matter which side is which variable.
   
   The variable C{\displaystyle C} is angle Y.
   
   So, your formula should look like this: c2=52+92−2(5)(9)cos89{\displaystyle c^{2}=5^{2}+9^{2}-2(5)(9)\cos {89}}. , Do this using a calculator’s cosine function.
   
   Simply type in the angle measurement, then hit the COS{\displaystyle COS} button.
   
   If you don't have a scientific calculator, you can find a cosine table online, such as the one found at the Physics Lab website.You can also simply type in "cosine x degrees" into Google, (substituting the angle for x), and the search engine will give back the calculation.
   
   For example, the cosine of 89 is about
   0.01745.
   
   So, plug this value into your formula: c2=52+92−2(5)(9)(0.01745){\displaystyle c^{2}=5^{2}+9^{2}-2(5)(9)(0.01745)}. , You are multiplying 2ab{\displaystyle 2ab} by the known angle’s cosine.
   
   For example:c2=52+92−2(5)(9)(0.01745){\displaystyle c^{2}=5^{2}+9^{2}-2(5)(9)(0.01745)}c2=52+92−1.5707{\displaystyle c^{2}=5^{2}+9^{2}-1.5707} , Remember that when you square a number, you multiply it by itself.
   
   Square the two numbers first, then add them.
   
   For example:c2=52+92−1.5707{\displaystyle c^{2}=5^{2}+9^{2}-1.5707}c2=25+81−1.5707{\displaystyle c^{2}=25+81-1.5707}c2=106−1.5707{\displaystyle c^{2}=106-1.5707} , This will give you the value of c2{\displaystyle c^{2}}.
   
   For example:c2=106−1.5707{\displaystyle c^{2}=106-1.5707}c2=104.4293{\displaystyle c^{2}=104.4293} , You will likely want to use a calculator for this step, because the number you are finding the square root of will have many decimal places.
   
   The square root is equal to the length of the missing side of the triangle.For example:c2=104.4293{\displaystyle c^{2}=104.4293}c2=104.4293{\displaystyle {\sqrt {c^{2}}}={\sqrt {104.4293}}}c=10.2191{\displaystyle c=10.2191}So, the missing side length, c{\displaystyle c}, is
   10.2191 cm long.
3. 3

   Step 3: Plug the known values into the formula.

4. 4

   Step 4: Find the cosine of the known angle.

5. 5

   Step 5: Complete the necessary multiplication.

6. 6

   Step 6: Add the squares of the known sides.

7. 7

   Step 7: Subtract the two values.

8. 8

   Step 8: Take the square root of the difference.

Detailed Guide

About the Author