The set of all possible values of in the interval (0,) for which the points (1,2) and (,) …

Question

The set of all possible values of $	heta$ in the interval $(0,\pi)$ for which the points $(1,2)$ and $(\sin	heta,\cos	heta)$ lie on the same side of the line $x+y=1$ is:

Accepted Answer

Let $f(x,y)=x+y-1$. Since $(1,2)$ and $(\sin\theta,\cos\theta)$ lie on the same side, $f(1,2)\cdot f(\sin\theta,\cos\theta)>0$:

$$2[\sin\theta+\cos\theta-1]>0 \implies \sin\theta+\cos\theta>1 \implies \sqrt2\sin\left(\theta+\tfrac\pi4\right)>1$$

$$\sin\left(\theta+\tfrac\pi4\right) > \frac1{\sqrt2} \implies \theta+\frac\pi4 \in \left(\frac\pi4,\frac{3\pi}4\right) \implies \theta\in\left(0,\frac\pi2\right)$$

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