Question

A triangle has sides A, B, and C. The angle between sides A and B is `(7pi)/12` and the angle between sides B and C is `pi/12`. If side B has a length of 12, what is the area of the triangle?

Answer 1

`A=24.053` `units^2`

Explanation:

`Acolor(white)(00)color(black)(pi/12)color(white)(0000000)acolor(white)(00)color(white)(0)`

`Bcolor(white)(00)color(white)(pi/1 3)color(white)(0000000)bcolor(white)(00)color(black)(12)`

`Ccolor(white)(00)color(black)((7pi)/12)color(white)(0000000)c color(white)(00)color(white)(0)`

Let's find the remaining angle, `B`:

`pi-pi/12-(7pi)/12` leaves us with `pi/3`

`Acolor(white)(00)color(black)(pi/12)color(white)(0000000)acolor(white)(00)color(white)(0)`

`Bcolor(white)(00)color(black)(pi/3)color(white)(.)color(white)(0000000)bcolor(white)(00)color(black)(12)`

`Ccolor(white)(00)color(black)((7pi)/12)color(white)(0000000)c color(white)(00)color(white)(0)`

Now we should use law of sines

`(sin(pi/3))/12=(sin(pi/12))/a`

`a~~3.59`

`(sin(pi/3))/12=(sin((7pi)/12))/c`

`c~~13.4`

Now, to find the area, we use the equation `A=(hxxb)/2`, where `h` is the height.

`color(white)(a)color(white)(- - - - - - - - - -)color(black)(/)color(black)(|)`
`color(white)(a)color(white)(- - - - - - - - -)color(black)(/)color(white)(-)color(black)(|)`
`color(white)(a)color(white)(- - - - - - - -)color(black)(/)color(white)(- - .)color(black)(|)`
`color(white)(a)color(white)(- - - - - - -)color(black)(/)color(white)(- - -0)color(black)(|)`
`color(white)(- - - -)color(black)(a)color(white)(00000)color(black)(/)color(white)(- - - -0)color(black)(|)color(black)(h)`
`color(white)(a)color(white)(- - - - -)color(black)(/)color(white)(- - - - -0)color(black)(|)`
`color(white)(a)color(white)(- - - -)color(black)(/)color(white)(- - - - - -0)color(black)(|)`
`color(white)(a)color(white)(- - -)color(black)(/)color(white)(- - - - - - -0)color(black)(|)`
`color(white)(a)color(white)(- -)color(black)(/)color(white)(- - - - - - - -0)color(black)(|)`
`color(white)(-)color(black)(/)color(black)(..)color(black)(B)color(black)(........................................)`

`a=3.59`, and the angle `B` is `pi/3`. We need to find the remaining length, `h`.

`sin(pi/3)=h/(3.59)`

`sin(pi/3)xx3.59=h`

`h=3.11`

Now we can find the area:

`A=(hxxb)/2`

`A=(3.59xx13.4)/2`

`A=24.053` `units^2`