The Cosine Rule

Cosine Rulevideo

Cosine rule

What is the cosine rule?

• The cosine rule is used in non right-angled triangles

  • It allows us to find missing side lengths or angles

• It states that for any triangle

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

• Where

  • $a$ is the side opposite angle A

  • $b$ and $c$ are the other two sides

    • $b$ and $c$ are either side of angle A

    • A is the angle between them

How do I use the cosine rule to find a missing length?

• Use the cosine rule for lengths

  • when you have two sides and the angle between them

  • and you want to find the opposite side, a

• Start by labelling your triangle with the angles and sides

  • Angles have upper case letters

  • Sides opposite the angles have the equivalent lower case letter

• Substitute values into $a^2=b^2+c^2-2bc \cos A$

  • Make a the subject (don't forget to square root)

How do I use the cosine rule to find a missing angle?

• Use the cosine rule for angles

  • when you have all three sides

  • and you want to find an angle

• It helps to rearrange the formula as follows, by adding $2bc \cos A$ to both sides then making $\cos A$ the subject

$\begin{array}{rcl}{a}^2& =& b^2+c^2-2bc \cos A\\ a^2+2bc \cos A& =& b^2+c^2\\ 2bc \cos A& =& b^2+c^2-a^2\\ \cos A& =& \frac{b^2+c^2-a^2}{2bc}\end{array}$

• Use the formula $\begin{array}{rcl}\cos A& =& \frac{b^2+c^2-a^2}{2bc}\end{array}$ to find the unknown angle A

  • Remember, A is the angle between sides b and c

    • (you may need to relabel the triangle)

  • You will need to use inverse cosine at the end, ${\cos }^{-1}\left(...\right)$

• Unlike the sine rule, there is no ambiguous case of the cosine rule