A triangle has sides A,B, and C. If the angle between sides A and B is (pi)/8π8, the angle between sides B and C is pi/6π6, and the length of B is 13, what is the area of the triangle?

2 Answers
Dean R.
Apr 23, 2018

text{area} approx 20.3798 area≈20.3798

Explanation:

First of all, let's rewrite in standard notation. It really helps to do these if we always label the triangle consistently.

The triangle has sides a,b,ca,b,c (small letters) where b=13b=13 with opposing angles C=pi/8=22.5^circC=π8=22.5∘, A=pi/6=30^circA=π6=30∘.

The remaining angle is

B=pi - A-C= pi-pi/8-pi/6 = { 17pi }/24 = 127.5^circ B=π−A−C=π−π8−π6=17π24=127.5∘

The Law of Sines says

b/sin B = c/sin CbsinB=csinC

c = { b sin C }/ sin B = {13 sin(22.5^circ) }/ sin(127.5^circ )c=bsinCsinB=13sin(22.5∘)sin(127.5∘)

Now the area is

text{area} = 1/2 bc sin A = frac{ (13)^2 sin(30^circ) sin(22.5^circ) } {2 \ sin 127.5^circ }

I'm tempted to work out an exact answer, because those sines are all expressible using the usual operations including square root. But it's late, and the calculator says

text{area} approx 20.3798

sankarankalyanam
Jun 18, 2018

color(maroon)(A_t == 20.37 " sq units"

Explanation:

b = 13, hat A = pi/6, hat C = pi/8, hat B = pi - pi/6 - pi/8 = (17pi)/24

Applying the Law of sines,

a = (b sin A) / sin B = (13 * sin (pi/6)) / sin ((17pi)/24) ~~ 8.19

Formula for area of " Delta, A_t = (1/2) * a b * sin C#

color(maroon)(A_t = (1/2) * 8.19 * 13 * sin ( pi/8) = 20.37 " sq units"

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