A triangle has sides A, B, and C. Sides A and B have lengths of 5 and 3, respectively. The angle between A and C is (19pi)/2419π24 and the angle between B and C is (pi)/8π8. What is the area of the triangle?

1 Answer
Sep 28, 2016

A ~~ 1.94 units^2A1.94units2

Explanation:

Let's use the standard notation where the lengths of the sides are the lowercase letters, a, b, and c and the angles opposite the sides are the corresponding uppercase letters, A, B, and C.

We are given a = 5, b = 3, A = (19pi)/24, and B = pi/8a=5,b=3,A=19π24,andB=π8

We can compute angle C:

(24pi)/24 - (19pi)/24 - (3pi)/24 = (2pi)/24 = pi/1224π2419π243π24=2π24=π12

We can compute the length of side c using either the law of sines or the law of cosines. Let's use the law of cosines, because it does not have the ambiguous case problem that the law of sines has:

c² = a² + b² - 2(a)(b)cos(C)

c² = 5² + 3² - 2(5)(3)cos(pi/12)

c = sqrt(5.02)

Now we can use Heron's Formula to compute the area:

Correction made to the following lines:

p = (5 + 3 + sqrt5.02)/2 ~~ 5.12

A = sqrt(5.12(5.12 - 5)(5.122 - 3)(5.12 - sqrt5.02)

A ~~ 1.94