Question

A triangle has sides A, B, and C. The angle between sides A and B is `pi/12` and the angle between sides B and C is `pi/12`. If side B has a length of 3, what is the area of the triangle?

Answer 1

Area `= 9/(4(2+sqrt(3)))`

Explanation:

(see diagram)

By Half Angle Formula for `tan`
`color(white)("XXX")tan(pi/12) = sin(pi/6)/(1+cos(pi/6))`
using standard values for `sin` and `cos`
`color(white)("XXX")tan(pi/12)= 1/(2+sqrt(3))`

The height of the triangle is
`color(white)("XXX")h=3/2 xx tan(pi/12)`

and the area is
`color(white)("XXX")("base " xx " height")/2`

`color(white)("XXX")=(3xx3/2xxtan(pi/2))/2`

`color(white)("XXX")=9/4xx1/(2+sqrt(3))`