Getting an acute angle for an obtuse angle using law of Sines.

Question

I have done this problem over and over again. I even looked up tutorials on how to properly use law of sines. It's rather embarrassing that I'm struggling so much wish this simple trigonometric stuff.

Here's the picture of the triangle. I'm trying to solve for angle $\angle{C}$. Angle $\angle{C}$ is definitely supposed to be obtuse.

I keep getting: $$\dfrac{\sin(21.55)}{7.7} = \dfrac{\sin(C)}{16}$$

I simplify and take the $\arcsin\left(16 \cdot \dfrac{\sin(21.55)}{7.7}\right)$

And I can't get an obtuse angle. Anyone know why?

Answer 1

Typically, the range of $\arcsin(x)$ is $[-\pi/2,\pi/2]$. This thereby eliminates the obtuse angle you want. To get the obtuse angle you want, all you need to do is to realize that $$\sin(\pi - \alpha) = \sin(\alpha)$$ Hence, $180^{\circ}- \arcsin(16 \sin(21.55^\circ)/7.7)$ should give you the answer you need.

Answer 2

Because of the range of arcsin you answer is the supplement to the correct answer. In order to avoid this difficulty always look for angles based on the side lengths in increasing order. In other words if you have a choice as to which angle to solve for first always choose the angle opposite the shortest of the two sides you have.

In this problem you are forced to find the angle C first so you must recognize that it is obtuse and therefore the answer you get is the supplement.