# Non-right angled triangles

### The cosine rule

$a^2=b^2+c^2-2bc\cos(A)$
$b^2=a^2+c^2-2ac\cos(B)$
$c^2=a^2+b^2-2ab\cos(C)$

### The sine rule

$\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}$

### The area of a non-right angled triangle

$\frac{1}{2}ab\sin(C)$

Note that to maintain accuracy, store values on your calculator and leave rounding until the end of the question. You can round when jotting down working but you should retain accuracy throughout calculations.

Example 1
x can be found using the cosine rule according to the labels in the triangle on the left. Choose a=3, b=5, c=x and so C=70:
$x^2=3^2+5^2-2\times3\times 5\times \cos(70)=9+25-10.26=23.74$ It follows that x=4.87 to 2 decimal places.
The area of this triangle can easily be found by substituting a=3, b=5 and C=70 into the formula for the area of a triangle.
$\text{Area }=\frac{1}{2}\times 3\times 5\times \sin(70)=7.05$
to 2 decimal places.

Example 2

x can be found using the sine rule according to the labels in the triangle on the left. Choose a=2.1, b=3.6 and so A=x and B=50. We only need to use one of the equations in the sine rule:

$\frac{2.1}{\sin(x)}=\frac{3.6}{\sin(50)}=4.699\Longrightarrow 2.1=4.699\sin(x)$
$\Longrightarrow \sin(x)=\frac{2.1}{4.699}=0.447$.
It follows that
$x=\sin^{-1}(0.447)=26.542$
to 3 decimal places.