Question

Solve the following spherical triangle using Law of Sine and Cosine

A= 137°28´, C=135°44´, c= 160° Find : a

Answer 1

$A = 137\degree28\rq = 137.47\degree$

$С = 135\degree44\rq = 135.73\degree$

Using Law of Sine and Cosine for spherical triangle $\frac{sin (A)}{sin (a)} = \frac{sin (B)}{sin (b)} = \frac{sin (C)}{sin (c)}$,

hence $\frac{sin (A)}{sin (a)} = \frac{sin (C)}{sin (c)}$.

According to the task $\frac{sin (137.47\degree)}{sin (a)} = \frac{sin (135.73\degree)}{sin (160\degree)}$ .

Then $sin (a) = \frac{sin (137.47\degree) * sin (160\degree)}{sin (135.73\degree)};$

$sin (a) = \frac{0.68 * 0.34 }{0.7};$

$sin (a) = 0.33$

Therefore $a = 19.3\degree$.

However $19.3\degree < 90\degree$, so the sine ambiguity appears (if angle $A > 90\degree$ (given), then side a should be $> 90\degree$).

That is why $a = 180\degree - 19.3\degree = 160.7\degree = 160\degree42\rq.$

Answer: $160\degree42\rq$