A triangle has sides A, B, and C. Sides A and B have lengths of 6 and 4, respectively. The angle between A and C is #(7pi)/24# and the angle between B and C is # (13pi)/24#. What is the area of the triangle?

1 Answer
May 2, 2016

Let us plot the triangle, and let us find its area using trigonometry.


First of all, our triangle is similar to the next one:

own source

Although there are many ways to find the area of a triangle (you can check it on Wikipedia ), we will use trigonometry:


The area of the above triangle can be calculated by:

#A = 1/2 a b sin gamma#

i.e. we must multiply two sides and the sine of the angle between them.

We know A and B sides, but we do not know the angle. We may calculate it by knowing that, for any triangle with angles #alpha, beta, gamma#:

#alpha + beta + gamma = pi rightarrow#

#rightarrow {7 pi}/24 + {13 pi}/24 + gamma = pi rightarrow#

#rightarrow gamma = {4 pi}/24 = pi/6#

And now:

#A = 1/2 cdot "A" cdot "B" cdot sin gamma = 1/2 cdot 6 cdot 4 cdot sin (pi/6) = color(blue) 6#