$$\cos x + \sin x = {1 \over 2}$$

$$ \Rightarrow {\left( {\cos x + {\mathop{\rm sinx}\nolimits} } \right)^2} = {1 \over 4}$$

$$ \Rightarrow {\cos ^2}x + {\sin ^2}x + 2\cos x\sin x = {1 \over 4}$$
$$\left[ \because {{{\cos }^2}x + {{\sin }^2}x = 1\, \,and \,\,2\cos x\sin x = \sin 2x} \right]$$

$$ \Rightarrow 1 + \sin 2x = {1 \over 4}$$

$$ \Rightarrow \sin 2x = - {3 \over 4},$$ so $$x$$ is obtuse and

$${{2\tan x} \over {1 + {{\tan }^2}x}} = - {3 \over 4}$$

$$ \Rightarrow 3{\tan ^2}x + 8\tan x + 3 = 0$$

$$\therefore$$ $$\tan x = {{ - 8 \pm \sqrt {64 - 36} } \over 6}$$

$$ = {{ - 4 \pm \sqrt 7 } \over 3}$$

as $$\tan x < 0\,$$

$$\therefore$$ $$\tan x = {{ - 4 - \sqrt 7 } \over 3}$$

$$2{\sin ^2}x + 5\sin x - 3 = 0$$

$$ \Rightarrow \left( {\sin x + 3} \right)\left( {2\sin x - 1} \right) = 0$$

$$\sin x = {1 \over 2}$$ and $$\,\,\sin x \ne - 3$$

Given that $$x \in \left[ {0,3\pi } \right]$$

So possible values of x are $$30^\circ $$, $$150^\circ $$, $$390^\circ $$, $$510^\circ $$. That means x have 4 values.

Concept : If a line makes the angle $$\alpha ,\beta ,\gamma $$ with x, y, z axis respectively then $$${\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma = 1$$$

In this question given that the line makes angle θ with x and z-axis and β with y−axis.

$$\therefore\: cos^2\theta+cos^2\beta+cos^2\theta=1$$

$$\Rightarrow\:2cos^2\theta=1-cos^2\beta$$

$$ \Rightarrow 2{\cos ^2}\theta = {\sin ^2}\beta $$

But given that $$sin^2\beta=3sin^2\theta$$

$$\therefore$$ $$2{\cos ^2}\theta = 3{\sin ^2}\theta $$

$$ \Rightarrow 2{\cos ^2}\theta = 3\left( {1 - {{\cos }^2}\theta } \right)$$

$$ \Rightarrow 2{\cos ^2}\theta = 3 - 3{\cos ^2}\theta $$

$$ \Rightarrow 5{\cos ^2}\theta = 3$$

$$ \Rightarrow {\cos ^2}\theta = {3 \over 5}$$