Question

A triangle has sides A, B, and C. The angle between sides A and B is `pi/3`. If side C has a length of `8` and the angle between sides B and C is `pi/12`, what is the length of side A?

Answer 1

`Acolor(white)(00)pi/12color(white)(00000)acolor(white)(00)color(red)(2.391)`

`Bcolor(white)(00)color(black)((7pi)/12)color(white)(00000)bcolor(white)(00)8.923`

`Ccolor(white)(00)pi/3color(white)(0)color(white)(00000)c color(white)(00)8`

Explanation:

Let's write this information in a table, where capital letters correspond to angles , lowercase to lengths :

`Acolor(white)(00)pi/12color(white)(00000)acolor(white)(00)`

`Bcolor(white)(00)color(white)(pi/12)color(white)(00000)bcolor(white)(00)`

`Ccolor(white)(00)pi/3color(white)(0)color(white)(00000)c color(white)(00)8`

All angles in a triangle add up to `180^o`, or `pi`

`pi/12+pi/3` only equals `(5pi)/12`.

`(12pi)/12-(5pi)/12=(7pi)/12`

That's the remaining angle:

`Acolor(white)(00)pi/12color(white)(00000)acolor(white)(00)`

`Bcolor(white)(00)color(black)((7pi)/12)color(white)(00000)bcolor(white)(00)`

`Ccolor(white)(00)pi/3color(white)(0)color(white)(00000)c color(white)(00)8`

We can use law of sines to find the other lengths:

`(sin(pi/3))/8=(sin(pi/12))/a`

`a~~2.391`

Just, for fun, let's also find length b

`(sin(pi/3))/8=(sin((7pi)/12))/b`

`b~~8.923`