Question

A triangle has sides A, B, and C. The angle between sides A and B is `pi/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?

Answer 1

Area `~~ 2.2877`

Explanation:

If `/_BC=pi/12` and `/_AB=pi/6`
then
`color(white)("XXX")/_AC = pi- (pi/12+pi/6) =(3pi)/4`

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By the Law of Sines:
`color(white)("XXX")A/sin(/_BC)=B/sin(/_AC)=C/sin(/_AB)`

With the given values:
`color(white)("XXX")A/sin(pi/12)=5/sin((3pi)/4)=C/sin(pi/6)`

So
`color(white)("XXX")A=5/sin((3pi)/4)*sin(pi/12)~~1.830127019`
and
`color(white)("XXX")C=5/sin((3pi)/4)*sin(pi/6)~~3.535533906`

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The semi-perimeter of the triangle, `s`, is
`color(white)("XXX")s=(A+B+C)/2~~5.182830462`

By Heron's Formula, the Area of the Triangle is
`color(white)("XXX")A=sqrt(s(s-A)(s-B)(s-C))`

`color(white)("XXXX")~~2.287658774`