# For how many values of $\theta$ such that $0<\theta<360$ do we have $\cos \theta = 0.1$?

For how many values of $\theta$ such that $0<\theta<360$ do we have $\cos \theta = 0.1$? (Note that $\theta$ is a measure in radians, not degrees!)

The period of $\cos(x)$ is $2\pi,$ $114pi = 358.14,$ so $\cos(x)$ repeats $114/2 = 57$ times. So the answer is $57 \cdot 2 = 114,$ but since cosine is split over y axis on the first period, we add $1$ to $114,$ and get $115.$
Is my answer correct? Thanks for any confirmation or correction!

• For that range of $\;\theta\;$ you give a complete turn around the trigonometric circle (except the end points which we don't care now about) . How many times exactly is every possible value for $\;\cos\theta\;$ attained while turning around? – DonAntonio Oct 29 '16 at 21:45
• @DonAntonio Twice, since there are $2\pi$ radians in each rotation. And also, is $115$ the correct asnwer to the problem? Thanks! – Yuna Kun Oct 29 '16 at 22:09
• Exactly: only twice. As you seem to be using degrees, I get that happens at $\;84.26^\circ\;,\;\;360-89.43=275.74^\circ\;$ degrees. In radians I get $\;\pm 1.47\;$ radians. – DonAntonio Oct 30 '16 at 10:07

Well my calculator says $\arccos 0,1 = 1,4706289... + 57 \cdot 2\pi = 359,61...\$ so you are right.