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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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Area & Perimeter Transformations Coordinate Geometry 31.0 TRIGONOMETRY 31.1 SIN / COS / TAN 31.2 SINE RULE
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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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