# 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

#### Explanation:

Our goal will be to use

**Step 1:** Find the value of

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

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# .