Question

Find the set of values of $\alpha$ for which the point $(\sin \alpha ,\cos \alpha )$ does not lie outside the curve $2y^2+x-2=0$ in the interval $[\frac{\pi }{2},\frac{3\pi }{2}]$

Answer 1

For a point $(x_1,y_1)$ to lies inside the curve $2y^2+x-2=0$, it must satisfy
$2y_1^2+x_1-2\le 0$
Here $x_1=sin\alpha$ and $y_1=cos\alpha$
Therefore
$2{\sin }^2\alpha +\cos \alpha -2\le 0$
$2-2{\cos }^2\alpha +\cos \alpha -2\le 0$
$\cos \alpha (1-2\cos \alpha )\le 0$
But
In the interval of $[\frac{\pi }{2},\frac{3\pi }{2}]$
$\cos \alpha \le 0$.
Therefore in the above inequality
$1-2\cos \alpha \ge 0$
$2\cos \alpha -1\le 0$
$\cos \alpha \le \frac{1}{2}$
Therefore $\alpha$ can take any value in the given interval of $[\frac{\pi }{2},\frac{3\pi }{2}]$.