Deciding the Trig Rule (Edexcel IGCSE Maths B): Revision Note

Look at the number of marks for a question - if it is a lot, you are likely to need more than one trig rule!

Find the area of the triangle below.

Answer:

The area of a triangle can be found using the formula $A=\frac{1}{2}ab\sin C$

The three side lengths are known , but we need to find an angle in order to calculate the area
Because we know all three sides, any of the angles could be found

Find angle ABC using the cosine rule

Cosine Rule:  $a^2=b^2+c^2-2bc\cos A$,
where $A$ is the angle opposite side $a$  

$\begin{array}{rcl}A{C}^2& =& A{B}^2+B{C}^2-2\left(AB\right)\left(BC\right)\cos ABC\\ 4.4^2& =& 7.4^2+4.8^2-2\left(7.4\right)\left(4.8\right)\cos ABC\end{array}$

Rearrange to make$\cos ABC$ the subject

$\begin{array}{rcl}4.4^2-\left(7.4^2+4.8^2\right)& =& -2\left(7.4\right)\left(4.8\right)\cos ABC\\ \cos ABC& =& \frac{4.4^2-7.4^2-4.8^2}{-2\left(7.4\right)\left(4.8\right)}\end{array}$

Use your calculator to find the value of $\cos ABC$

$\begin{array}{rcl}\cos ABC& =& \frac{487}{592}\end{array}$

Use the cos^-1 button on your calculator to find the value of $ABC$

$\begin{array}{rcl}ABC& =& {\cos }^{-1}\left(\frac{487}{592}\right)\\ & =& 34.65054...\end{array}$

Now we can find the area of the triangle using the formula and angle ABC as the known angle

$\begin{array}{rcl}\mathrm{Area}& =& \frac{1}{2}\times 4.8\times 7.4\times \sin 34.65054...\\ & =& 10.09779...\end{array}$

$\mathrm{Area}=10.1{\mathrm{cm}}^2$ (3.s.f)