First slide

Equation of line in Straight lines

Question

Let $P (\sin θ, \cos θ), (0 \leq θ \leq 2 π)$ , be a point in a triangle with vertices $(0, 0), (\sqrt{3 / 2}, 0), and (0, \sqrt{3 / 2})$ . Then,

Moderate

A

$0 < θ < π / 12$
B

$5 π / 2 < θ < π / 2$
C

$0 < θ < 5 π / 2$
D

$5 π / 2 < θ < π$

Solution

The equations of lines along OA, OB, and AB are y= 0, x= 0, and x + y $= \sqrt{3 / 2}$ , respectively.
Now, P and B will lie on the same side of y: 0 if cos 0 > 0. Similarly,
P and A will lie on the same side of x=0 if sin $θ$ > 0, and P and O will lie on the same side of x + y $= \sqrt{3 / 2}$
if $\sin θ + \cos θ < \sqrt{3 / 2}$
Hence, P will lie inside $ΔABC if \sin θ > 0, \cos θ > 0$ and
$\sin θ + \cos θ < \sqrt{3 / 2}$ now,
$\sin θ + \cos θ < \sqrt{\frac{3}{2}}$
now $\sin (θ + \frac{π}{4}) < \sqrt{\frac{3}{4}}$
Since $\sin θ > 0 and \cos θ > 0, 0 < θ < π / 12 or 5 π / 12 < θ < π / 2$

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