Confused about $\csc(x)$ The title says it all, let's take this simple example:

I want to find $c$.
Using $\sin$ I get:


*

*$\sin(30)$ = $4 / c$

*$c = 4 / \sin(30) = 8$
Now, using $\csc$:


*

*$\csc(30) = c / 4$

*$c = \csc(30) * 4 = 8$
But also:


*

*$\csc(30) = c / 4$

*$\tan(30) = 4 / a$

*$4 = \tan(30) * a$

*$c = \csc(30)\cdot \tan(30)\cdot c = c\cdot (c / 4)\cdot (4 / a) = c$

*OR: $\csc(30) \cdot \tan(30)\cdot c = \sec(30)\cdot  c \approx c$
What's gone wrong here, I assumed $c$ would cancel out?
EDIT 1: Why isn't $c$ cancelling out in the last example, even after the fix?
EDIT 2: Fixed steps 2 and 3 (changed $c$ to $a$). Thank you for your help, silly mistakes.

$$\sec30^{\circ}=\frac{1}{\cos30^{\circ}}=\frac{2}{\sqrt3}.$$
