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Faris3 · Nov 7, 2011, 12:15 PM

Hello there. I was having some trouble with a little question.
If cos(x)= 1/2 then x = arcos(1/2) which is pie/3.
Now if cos(x)= -1/2, is x=arcos(-1/2) still pie/3 or is arcos(-1/2) = 2pie/3?
Or are maybe both of them right? Thanks in advance.

ebaines · Nov 7, 2011, 12:29 PM

You are correct that if $\cos(x) = 1/2$ then $x = \frac 1 3 \pi.$ , although it could also equal $\frac 5 3 \pi$ , because $\cos (\pi/3) = \cos (5 \pi/3) = 1/2.$ . The complete set of solutions is:

$<br /> \arccos (1/2 ) = \left { \pi/3 \pm 2 \pi n\\<br /> \5 \pi/3 \pm 2 \pi n \right$ for $n$ equals any integer.

If cos(x) = -1/2 then $\arccos(1/2) = x = 2/3 \pi + 2 \pi n$ or $x = 4/3 \pi + 2 \pi n$ . It can't be $\pi/3$ because we already know that $\cos (\pi/3) = 1/2$ , not $-1/2$ .