Question

A triangle has sides A, B, and C. The angle between sides A and B is `(pi)/3` and the angle between sides B and C is `pi/6`. If side B has a length of 13, what is the area of the triangle?

Answer 1

`A_triangle=(169sqrt 3)/8 approx 36.59`.

Explanation:

Our goal will be to use `A_triangle = 1/2 a b sin C`. We know `b=13` and `angle C = pi/3`, so we need to find `a`.

Step 1: Find the value of `angle B`.

Using the fact that the sum of all 3 angles in a triangle is `pi`, we get

`angle A + angle B + angle C = pi`
`pi/6"  "+ angle B + pi / 3"  "= pi`
`"           "angle B "            "= pi/2`

So `angle B = pi/2`.

Step 2: Find the length of `a`.

We now use the sine law for triangles to get

`a/sinA=b/sinB`

`a/sin(pi/6)=13/sin(pi/2)`

`"      "a"      "=(13sin(pi/6))/sin(pi/2)`

`"      "a"      "=(13(1/2))/(1)=13/2`

So `a=13/2`.

Step 3: Find the area of the triangle.

We can now use the following formula for a triangle's area:

`A_triangle=1/2 a b sin C`

`A_triangle=1/2 * 13/2 * 13 * sin (pi/3)`

`A_triangle=169/4 * sqrt 3 / 2`

`A_triangle=(169sqrt 3)/8"     "approx 36.59`.