I am trying to find all of the angles $\theta$ between $0°$ and $180°$ that satisfy $\cos(\theta) = \frac {6} {7}$.
Here is what I did:
$\cos^{-1}$($\frac {6}{7}$), which gave me $31$. Then I did $180-31$, which is $149$. So my final answer was $31$ and $149$, but that is incorrect. What did I do wrong?
$\endgroup$ 24 Answers
$\begingroup$The only answer is $\arccos\left(\frac67\right)$, because $\cos(\theta)$ decreases as $\theta$ goes from $0^\circ$ to $180^\circ$ and therefore it cannot take the same value twice in that interval.
$\endgroup$ $\begingroup$If cos is >0 then the inverse angles lie either in the first or in the fourth quadrants. Since you include only the first and second, what remains is the angle in the first quadrant, i.e., $31^{\circ}.$
$\endgroup$ $\begingroup$For this question the answer is $31$ only because the next one would be $301$ which is outside the range. The one previous to $31$ is $-31$ which is outside the range again.
$\endgroup$ $\begingroup$Actually $\cos(π-x)=-\cos(x)$ so $149$ is not a solution.
$\endgroup$
