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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!

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2 Answers 2

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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 $\implies n = \pi- \sin^{-1} k$

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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. enter image description here

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  • $\begingroup$ I'm more of a formula type of guy, but this definitely helped me understand the concepts of the formulas so thank you for that! $\endgroup$
    – Whiteclaws
    Commented Feb 21, 2018 at 20:38
  • $\begingroup$ Memorizing formulas has an upper bound on how many you can keep track of, and know which to apply when. A lot of people do make it through trig with based on intensive memorization, and I know that some teachers teach it that way. But trig / pre-calc is about as far as you can get before the concepts become far more important that the formula. $\endgroup$
    – Doug M
    Commented Feb 21, 2018 at 20:46

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