CED = $\angle 75 ^\circ$,
CDE = $\angle 60 ^\circ$,
DCE = $\angle 45 ^\circ$,
Radius = 1.
The question ask to find lenght CD, DE, EC without trigonometric functions. What does it mean?
I tried using cosinus law, which is $CD = \sqrt{2r^2 -2r^2\cos 150^\circ}$ Does it mean i am using trigonometric functions?
Since $\measuredangle COE=120^{\circ}$, by the Pythagoras's theorem we obtain: $$CE=2\sqrt{1-\left(\frac{1}{2}\right)^2}=\sqrt3.$$ Also, since $\measuredangle DOE=90^{\circ},$ by the Pythagoras's theorem again we obtain: $$DE=\sqrt{1^2+1^2}=\sqrt2.$$ Now, let $EF$ be an altitude of the triangle.
Thus, since $\measuredangle FED=30^{\circ},$ we obtain $$FD=\frac{1}{2}\sqrt2=\frac{1}{\sqrt2}.$$ Also, since $\measuredangle ECF=45^{\circ},$ we obtain $$CF=\frac{\sqrt3}{\sqrt2}$$ and
$$CD=FD+CF=\frac{1+\sqrt3}{\sqrt2}.$$