Evaluate tan(Sin-1(-5/13)). Answer as a fraction please.

Question

zzr0ck3r · Answer

Sorry on tablet and the site won't let me draw

kinggeorge · Answer

There should be a particular triangle that has two sides of length 5 and 13. Can you tell me what the third side length is?

avanti · Answer

12 is the third side

kinggeorge · Answer

Right. So you can draw a triangle with side lengths 5,12,13. |dw:1375483336420:dw|And $\sin(x)=5/13$. Now can you tell me what $\tan(x)$ is?

avanti · Answer

tan(x)=5/12

kinggeorge · Answer

Bingo. Now we just need to fit these pieces together.$$\sin(x)=5/13\implies\sin^{-1}(5/13)=x\implies\sin^{-1}(-5/13)=-x$$Then we take the tangent.$$\tan(\sin^{-1}(-5/13))=\tan(-x)=-5/12$$All this make sense?

avanti · Answer

oh okay thank you! how come you made the 5/13 and the x negative in the third part of the  2nd row?

kinggeorge · Answer

If you look at your original problem, we want to find $\tan(\sin^{-1}(-5/13))$, which has a negative in front of the 5/13. So I made the 5/13 negative. Then, $\sin$ has the property where $\sin(-x)=-\sin(x)$. So I applied that to get the negative in front of the x.

avanti · Answer

okay that makes sense, thank you very much!

kinggeorge · Answer

You're very welcome.