# A triangle has sides A, B, and C. The angle between sides A and B is (pi)/4 and the angle between sides B and C is pi/6. If side B has a length of 12, what is the area of the triangle?

May 7, 2016

area of triangle is $\left(\frac{1}{2}\right)$(base)(height) = $\left(\frac{1}{2}\right) \cdot A \cdot B$
= $\left(\frac{1}{2}\right)$(12) ((12/(sin60π))*sin30º)

#### Explanation:

Working in degree, π = 180º,

The Angle between C and A is
(180º -90º -30º) = 60º

Hence using sine rule,

A/sin(angle between B&C) = B/sin(angle between C&A) =
C/sin(angle betwen A&B)

And given that $\angle b e t w e e n A B$ is a right angle,

you need to only find A where

A/sin(angle between B&C) =B/sin(angle between C&A),

A= (B/sin(angle between C&A))*sin(angle between B&C)

= (12/(sin60π))*sin30º

Hence, area of triangle is $\left(\frac{1}{2}\right)$(base)(height) = $\left(\frac{1}{2}\right) \cdot A \cdot B$
= $\left(\frac{1}{2}\right)$(12) ((12/(sin60π))*sin30º)