Specifically something like sinx=-4 and cosx=-1
I think your question is misleading. To find the "sine of a number" you would use a calculator. But from your examples, it seems you want to solve for x, when x is in a trig function.
Simply take the inverse of the trig function.
$\displaystyle
\cos{x} = \frac{1}{2}
$
$\displaystyle x = \arccos{\frac{1}{2}}$
$\displaystyle x = \frac{\pi}{3}$
The inverse functions are symbolized by the inverse sign, e.g., $\displaystyle sin^{-1}$.
Also, if you are not allowed to use a calculator, the you can use a "unit circle" for common angles.
I've read that you can use triangles to solve for inverse functions, but I can't wrap my head around it. I understand how to get $\displaystyle
\sin{x} = \frac{1}{2}
$
using a special right triangle, but how can you get sinx=-4 using that technique?
Are you using only real numbers or can you use imaginary numbers for this problem?
If you are using only real numbers, then sinx = -4 has no real solution, since -1 <= sinx <= 1. So the answer would be "No Solution."
As for the other example you gave, cosx = -1, that is in the domain of cos, so we can get x = pi.
Patrick
Two methods come to mind. My favorite: use a calculator. TI-84 Plus is my personal pick.
Of course, you can do this one by using the unit circle also. $\displaystyle \frac{\pi}{2}$ is in radians. In degrees, that same angle is 90 degrees. If on a xy plane you draw a line that is at 90 degrees, you are drawing a line straight up. One unit up is at the point (0,1), right? Since cos(t) = x, then we just look at the x value, which is zero. So...
$\displaystyle \cos{\frac{\pi}{2}} = 0$
Patrick
I was always told to memorize the unit circle. And I never, ever did. I was able to get by though by using my calculator and by creating the unit circle using simple logic whenever I really needed it. I still don't have most of the basic angles memorized, I always check with my calculator.
That being said, my advice is to memorize the common angles, as it will really help out in the years to come. I still waste time creating the unit circle when if I just had it memorized it would save me a minute or two.