# Cosine Rule

## Introduction

Cosine rule is another trigonometry rule that allows you to find missing sides and angles of triangles. This rule is used
when we know an angle in between two angles or when we know 3 sides of the triangle. There are several different
forms of this rule as you can see on the right. The top ones are for finding missing sides while the bottom ones are for
finding missing angles.
## Finding Missing Sides Using Cosine Rule

To use cosine rule we first have to annotate the sides and angles so
that A, B & C are the angles and a, b and c the sides opposite to the
corresponding angle. Then you have to use the correct rule. In this
case we want to find the missing side a which is why we use the
rule starting with a². Now simply substitute the values from the
triangle into the formula and then solve the individual parts. Pay
attention to the order of operation. Use your calculator to find a² and
then square root your answer to find the missing side.
## Finding Missing Angles Using Cosine Rule

In order to find a missing angle using cosine rule, we need to know
all the sides of the triangle. Again, it is wise to annotate the sides
and angles. In the example on the left we have to use the Cos B
rule since the missing angle is B. To solve the equation simply
substitute in the values and then simplify the equation until you are
left with a simple numerical fraction. Since we want to find B and
nod Cos B put this fraction into the inverse cosine function of your
calculator. This will give you the size of the unknown angle. In this
case, it is 35.5°. Have a look at a couple of examples underneath.
## Examples

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Example 1
In this example, we have used cosine rule to find the missing angle
A. We have substituted the values into the equation and simplified
it before using the inverse cosine function on the calculator to find
the missing angle.

Example 2
In this example, we have used cosine rule to find the missing side c
of the triangle. We have substituted the values into the equation and
simplified it before square rooting 451 to find the length of side c:
21.2 cm.

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Chapter 31.3: Learning Outcomes
Students will learn how to find missing angles of non right-angled
triangles using cosine rule!
Students will learn how to find missing sides of non right-angled triangles
using cosine rule!

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