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$\cos x = -1/2$ can occur in quadrants 2 or 3, that gives it 2 answers, however the answer sheet only shows one. Does this mean im doing something completely wrong, or are they just not showing the other one?

Thanks :D

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It depends on the exact wording of the question. There are exactly two answers if we specify that answers must be in the interval $[0,2\pi)$ ($[0, 360)$ in degrees). There are infinitely many answers if we make no specification, There may be an implicit restriction, such as "angle of a triangle," that restricts answers to the interval $(0,\pi)$. – André Nicolas Sep 4 '11 at 21:16
up vote 5 down vote accepted

It depends upon the range answers are allowed in. If you allow $[0,2\pi)$, there are certainly two answers as you say:$\frac{2\pi}{3}$ and $\frac{4\pi}{3}$. The range of arccos is often restricted to $[0,\pi)$ so there is a unique value. If $x$ can be any real, then you have an infinite number of solutions: $\frac{2\pi}{3}+2k\pi$ or $\frac{-2\pi}{3}+2k\pi$ for any integer $k$.

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If you take the unit circle interpretation: the points corresponding to angles of $\frac23\pi$ and $\frac43\pi$ on the unit circle do have the same x-coordinate. – J. M. Sep 4 '11 at 21:18

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