A triangle has sides A, B, and C. The angle between sides A and B is `(5pi)/6` and the angle between sides B and C is `pi/12`. If side B has a length of 5, what is the area of the triangle?

We first need to calculate the the third angle.

`B= pi -(pi/12 +(5pi)/6)=pi/12`

We can calculate the length of side `a` using the Sine Rule:

`sin(A)/a=sin(B)/b=sin(C)/c`

`sin(pi/12)/a=sin(pi/12)/5=>a = (5sin(pi/12))/(sin(pi/12))=5`

We can find the altitude using:

`a * sin(C)`

Since area is `1/2xxbase xxheight`

We have:

`1/2*5*5*sin((5pi)/6)= 6.25` `units^2`

`:.(cancel(5pi)^color(magenta)5)/cancel6^color(magenta)1xx180^color(magenta)30/cancelpi^color(magenta)1=150^@= angle` between sides A and B

`:.(cancelpi^color(magenta)1)/cancel12^color(magenta)1xxcancel180^color(magenta)15/cancelpi^color(magenta)1=15^@= angle` between sides B and C

`:.180-(150+15)=15^@= angle` between sides A and C

The `triangle`= a isoceles triangle

Area of `triangle=` `1/2` base`xx` height

`:.`Base`=C=C/(sin 150^@)=5/(sin15^@)`

multiply both sides by `sin150^@`

`:.C=(5xxsin150^@)/(sin15^@)`

`:.C=(5xx0.5)/0.258819045`

`:.C=(2.5)/0.258819045`

`color(magenta)(C=9.659=Base`

Perpendicular `heightcolor(magenta)(= D`

`:.D/5=Sin15^@`

multiply both sides by `5`

`:.D=5xxsin15^@`

`:.D=5xx0.258819045`

`:.color(magenta)(D=1.294 units` perpendicular height

`:.`The area of the triangle`color(magenta)(=1/2xx9.659xx1.294=6.249 units^2` to the nearest 3 decimal places