Finding each value for trigonometric inverse functions

Question

If I end up with $sin/cos/tan(n) = k$ , I can then solve for a value using each trigonometric function's inverse function, but that is only one of the values, that I can find... I looked up this subject, but I want confirmation on my notes:

let $ v \in \Bbb R: $

$sin (n) = k, n = sin^{-1}(k) \ and \ \ (\pi - sin^{-1}(k)) \pm 2\pi v $

$cos(n) = k, n = n = cos^{-1}(k) \ and \ \dots \pm 2\pi v$

$tan(n) = k, n = tan^{-1}(k) \pm \pi $

Correct me if I'm wrong, and do suggest a term to fill in the blanks, I'm going into trigonometry and I have doubts... Thank you very much!

Answer 1

According to the interval of definition and the range of trigonometric inverse functions we have that

• $\sin n = k, n = \sin^{-1} k \lor n = \pi-\sin^{-1} k $

• $\cos n= k, n = \cos^{-1} k \lor 2\pi -\cos^{-1} k$

• $\tan n = k, n = \tan^{-1} k \lor \pi +\tan^{-1} k $

Note all values are up to a multiple of $2\pi$.

Try to derive each one thinking to the definition and selecting a value for n in each quadrant.

For example $\sin^{-1}x$ is defined for $x\in[-1,1]$ and has range $[-\pi/2,\pi/2]$. Thus - if n is in the range $\implies n = \sin^{-1} k$ - if n is not in the range $\implies n = \pi- \sin^{-1} k$

Answer 2

I suggest you do not try to memorize a table.

Instead, hold a picture of something like this in your head about what these trig functions mean.