How to determine negative values of arc functions?

Question

Can someone explain me how do they work? I memorized the table of values for sin/cos/tan/cot and radian-degreee values. But, how come, when it is $\arcsin -1/2$ this is $-\pi/6$. So the same thing as $\arcsin 1/2$ just with a different sign. Why it is not $5\pi/6$? But when it is $\arccos -\sqrt{2}/2$ it is $3\pi/4$, not a $\pi/4$.

Answer 1

Hint : Consider $sin(-x)=-sin(x)$ and $cos(-x)=cos(x)$ for all real $x$.

The arcsin is usually defined as the inverse of the sine function in the interval [$-\frac{\pi}{2},\frac{\pi}{2}$], hence the negative values.

Answer 2

Another hint:

Make a sketch with the trigonometric circle. You'll see at once $\,\sin\frac{5\pi}6>0\,$ and $\,\cos\frac{3\pi}4<0$.

Answer 3

The arcsine function is defined as the inverse function of sin function but you should take into account the domain and the range of each. The domain of sin is $\left[ -\frac{\pi }{2},\ \frac{\pi }{2}\right] $ and its range is $% \left[ -1,\ 1\right] $ so the arcsine domain is $\left[ -1,1\right] $ and its range is $\left[ -\frac{\pi }{2},\ \frac{\pi }{2}\right] .$ So if you look for $\arcsin \frac{-1}{2}$ the answer should be $\frac{-\pi }{6}$ (Note that $\frac{5\pi }{6}=2.618$ is not in the range of arcsine function. Note that sin function is bijective from $\left[ -\frac{\pi }{2},\ \frac{\pi }{2}% \right] $ onto $\left[ -1,\ 1\right] .$ If you enlarge the domain to something greater than $\left[ -\frac{\pi }{2},\ \frac{\pi }{2}\right] $ the sin function wouldn't be one-to-one. (it remains onto but not one-to-one, not injective)

Answer 4

A very good and concise answer is already given by Idris but I want to give you a brief theory about them.

$\sin{^-1}$, $cos^{-1}$... $\cot^{-1}$ are actually relations unless we provide Domain and Range for them.

We say them relations because A relation is one that can create more than one output for a single input and a function can not have two or more outputs for a single input.

When we say that $\cos^{-1}{\frac{1}{2}}$, we are saying that find the angles whose cosine is $\frac{1}{2}$. There are infinitely many angles whose cosine is $\frac{1}{2}$ but since we want them as Functions, we restrict their domain and range and so in the range the output is $\frac{\pi}{3}$

The angle (output) we get from these inverse functions is called Principle angle and their notation is slightly changed i.e. we write the first letter with capital.

For instance,

$Cos^{-1}$, $Sin^{-1}$...$Cot^{-1}$.