Originally Posted by

**Grandad** Hello db5vryLook at the attached diagram, showing $\displaystyle \cos 60^o = 0.5$. The other angle in the diagram whose cosine is also $\displaystyle 0.5$ is in the fourth quadrant: it is $\displaystyle 360^o - 60^o = 300^o$.

If we allow angles of more than $\displaystyle 360^o$, the next angle whose cosine is 0.5 is $\displaystyle 360 + 60 = 420^o$; the next is $\displaystyle 660^o$, and so on.

So if $\displaystyle \cos 3x = 0.5$, the possible values of $\displaystyle 3x$ are $\displaystyle 60^o, 300^o, 420^o, 660^o, \dots$

To find the values of $\displaystyle x$, then, you'll need to divide each of these by $\displaystyle 3$ to get:

$\displaystyle x = 20^o, 100^o, 140^o, 220^o, \dots$

(Cosine is positive in the first and fourth quadrants, of course.)

Grandad