Cosine Rule

 

What is the Cosine Rule?

The cosine rule connects the sides and angles of any triangle, not just right-angled ones. It's especially useful when:

• You know two sides and the included angle, and want to find the third side.

• You know all three sides, and want to find one of the angles.

The general formula is:

$$a^2 = b^2 + c^2 - 2bc \cos A$$

Where:

• $a$ is the side opposite angle $A$

• $b$ and $c$ are the other two sides

• $A$ is the included angle between sides $b$ and $c$

 

 

Finding a Missing Side

Use the cosine rule when you know:

• Two sides of a triangle

• The angle between them

Steps:

1. Label your triangle carefully: angle $A$ should be between the known sides $b$ and $c$, with side $a$ opposite it.

2. Substitute into the formula:

$$a^2 = b^2 + c^2 - 2bc \cos A$$

3. Square root your final result to find $a$

Example:

Given:  $b = 6 \text{ cm},\ c = 9 \text{ cm},\ A = 60\degree$

Find $a$

 

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$$a^2 = 6^2 + 9^2 - 2 \cdot 6 \cdot 9 \cdot \cos(60\degree) \\ a^2 = 36 + 81 - 108 \cdot 0.5 = 117 - 54 = 63 \\ a = \sqrt{63} \approx 7.94 \text{ cm}$$

 

Finding a Missing Angle

Use this form of the cosine rule if you know all three sides and want to calculate an angle:

$$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$

Steps:

1. Identify the angle you need ($A$) and label the opposite side as $a$

2. Plug in the side lengths into the rearranged cosine rule

3. Use inverse cosine ($\cos^{-1}$) to calculate the angle

Example:

Given: $a = 7,\ b = 5,\ c = 6$

Find angle $A$

 

 

$$\cos A = \frac{5^2 + 6^2 - 7^2}{2 \cdot 5 \cdot 6} = \frac{25 + 36 - 49}{60} = \frac{12}{60} = 0.2 \\ A = \cos^{-1}(0.2) \approx 78.5\degree$$

 

 

 

 

Worked Example: Finding an Angle

A triangle has sides of length $PQ = 11 \text{ km},\ QR = 5.5 \text{ km},\ PR = 13.2 \text{ km}$. Find angle $Q$ to 1 decimal place.

 

 

We know all three sides, so use:

$$\cos B = \frac{4.2^2 + 3.8^2 - 7.1^2}{2 \cdot 4.2 \cdot 3.8} \\ \cos B = \frac{17.64 + 14.44 - 50.41}{31.92} = \frac{-18.33}{31.92} \approx -0.574 \\ B = \cos^{-1}(-0.574) \approx 125.0\degree$$

Final Answer: $125.0\degree (1 d.p.)$