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?

1 Answer
Dec 26, 2016

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.