Question

A triangle has sides A, B, and C. If the angle between sides A and B is `(5pi)/8`, the angle between sides B and C is `(pi)/3`, and the length of B is 16, what is the area of the triangle?

Answer 1

`784.42` square units (2 d.p.)

Explanation:

`A = 1/2ab sin C`
this formula can be used when working with `2` sides and `1` angle - `a` and `b` are the two sides that are not opposite the angle we are using.

however, there is only `1` side given, so we need to find a second side - either `a` or `c`.

first, find the angle between `a` and `c`:

`Sigma_angle(triangle) = pi`

angle between `a` and `c = pi - ((5pi)/8) - (pi/3)`

`= pi/24`

sine rule: `a/sin A = b/sin B = c/sin C`

`16/sin(pi/24) = a/sin(pi/3)`

`16sin(pi/3)=asin(pi/24)`

`16xx0.86603=0.13053a`

`a=(16xx0.86603)/0.13053=`

`a = 106.16, b = 16`

angle opposite `c =(5pi)/8`

`"Area"=1/2(106.16*16)sin(5pi)/8`

`=849.028xx0.9239`

`"Area" = 784.42` square units (2 d.p.)