Oblique Triangles

The basic trig functions are great and super useful. However they can not be applied directly to triangles that are not right-triangles. We can however break down ANY triangle into two right triangles and then apply basic trig. If we do this and then generalize we get the so called cosine rule and sine rule (or in American English the Law of Cosines and Law of Sines).

### Setup

For everything below we are going to use the following triangle. Note that the angles are capitals and the sides are lower case. Also the notice that angle A is opposite side a.

### Cosine Rule

To use the cosine rule you need to know all three sides OR two sides and one angle. The formula is given in two forms in the data booklet. They are the same just rearranged a bit:

(1)
\begin{align} c^2=a^2+b^2-2ab \; cos C \end{align}
(2)
\begin{align} cos C = \frac{a^2+b^2-c^2}{2ab} \end{align}

### Sine Rule

To use the sine rule you need to know either two angles and one side opposite either of those angles OR two sides and one angle opposite either of those sides.

(3)
\begin{align} \frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC} \end{align}

### Area of Oblique Triangles

For oblique triangles knowing the base and the height require some algebra. So a simple formula was developed that only requires the user to know the length of two sides and the angle between those sides.

(4)
\begin{align} Area=\frac{1}{2} ab \; sin C \end{align}

### Example

The following diagram shows triangle ABC.

Where $AB = 7 cm$, $BC = 9 cm$ and $A\hat{B}C=120^\circ$.

a) Find $AC$.
b) Find $B\hat{A}C$.