(sin θ, cos θ) and (3, 2) lies on the same side of the line x + y = 1, then θ lies between

Question

(sin θ, cos θ) and (3, 2) lies on the same side of the line x + y = 1, then θ lies between (A) (0, π /2) (B) (0, π ) (C) (π /4, π /2) (D) (0, π /4)

Tardigrade Admin · Accepted Answer

As (sinθ, cosθ) and (3, 2) lie on the same side of x + y - 1 = 0, they should be of same sign. ∴ sinθ + cosθ - 1 > 0 as 3 + 2 - 1 > 0 ⇒ √2sin(θ+(π/4)) >1 ⇒ sin(θ+(π /4)) > (1/√2) ⇒ 0 < θ < (π/4)

Q. $(s in θ, cos θ)$ and $(3, 2)$ lies on the same side of the line $x + y = 1$ , then $θ$ lies between

$(0, π /2)$

$(0, π)$

$(π /4, π /2)$

$(0, π /4)$

As $(s in θ, cos θ)$ and $(3, 2)$ lie on the same side of
$x + y - 1 = 0$ , they should be of same sign.
$∴ s in θ + cos θ - 1 > 0$ as $3 + 2 - 1 > 0$
$\Rightarrow 2 s in (θ + \frac{π}{4}) > 1$
$\Rightarrow s in (θ + \frac{π}{4}) > \frac{1}{2} \Rightarrow 0 < θ < \frac{π}{4}$

Q. (sinθ,cosθ) and (3,2) lies on the same side of the line x+y=1, then θ lies between

(0,π/2)

(0,π)

(π/4,π/2)

(0,π/4)