Mastering the Law of Sines: Troubleshooting Trig Problems

[USN2ENG]

Homework Statement

So, I am brushing up on my trig...and now I feel a bit slow.
I am attempting to use the law of sines for the attached problem and keep getting the wrong angle. For some reason I never learned the law of sines/cosines before today...

Homework Equations

A/Sin(a) = B/Sin(b) = C/Sin(c)

The Attempt at a Solution

4/sin(27)=5/sin(theta) then...
arcsin(5*sin(27)/4) = theta = 34.6

I know this is the angle beside it, but I don't know what I am doing wrong. Any help would be great!

 

[Bohrok]

The downfall of using the law of sines is that you can get the wrong answer when the angle is obtuse as you can see from the picture. Do you know how you can fix the "answer" of 34.58°?

 

[micromass]

The problem is that the arcsine doesn't give the value you want.

Basically, what arcsine does is it takes a value x in [-1,1] and it gives you an angle $\theta\in [-90^\circ, 90^\circ]$ such that $\sin(\theta)=x$.

But this $\theta$ isn't the only angle such that $\sin(\theta)=x$! Indeed $\pi-\theta$ is also such an angle. Unfortunately, the arcsine can only give you one answer instead of all the angles that satisfy (otherwise, your calculator would have to give you infinitely many values) and it chooses to ignore $\pi-\theta$.

The user should know that if he gets $\theta$ as an answer, that $\pi-\theta$ is also a good answer to the question and that $\theta$ might be the wrong answer. This is indeed the case here.

 

[USN2ENG]

Ahhh, ok, so apparently I need to use some logic too...haha. Thank you so much for the help. I was able to solve it by solving for the right triangle and then solving for the attached triangle, but this method tripped me up, unfortunately. Thank you all again for the explanation.