Question

A triangle has sides A, B, and C. The angle between sides A and B is `(5pi)/6`. If side C has a length of `2` and the angle between sides B and C is `pi/12`, what are the lengths of sides A and B?

Answer 1

A = B = 1.035

Explanation:

This one is deceptively easy, I think.

You know that the interior angles of a triangle add up to `pi` radians (or 180 degrees). You are given 2 of the angles, so you can calculate the angle between sides A and C = `pi - (pi/12 + (5pi)/6)`

...this comes out to `pi/12`. This is the same as the angle between sides B & C.

So you know you have an isosceles triangle, and therefore sides A and B are equal.

From your equilateral triangle, you can create two congruent right triangles, with hypotenuse A (or B), and base of length `C_1` and `C_2`. (`C_1 + C_2 = 2`, and since the two right triangles are equal, `C_1 = C_2`, so `C_1 = C_2 = 1`)

So now, using just a little trig, we know: `C_1/A = cos(pi/12)`
...since `C_1 = 1`, we have `1/A = cos(pi/12)`
Therefore, `A = 1/cos(pi/12)`

`A = 1.035` (rounding)