Question

If $x=\sin 1+\cos 1,y=\sqrt{\sin 1}+\sqrt{\cos 1}$ and $z=\sin 1-\cos 1,$ then which among the following relation between $x,y,z$ is correct ?

• A. $x>y>z$
• B. $y>x>z$
• C. $z>x>y$
• D. $z>y>x$

Answer 1

The correct option is B $y>x>z$

We know that, $1\text{radian}\approx {57}^{\circ }{18}^{\prime }\in (0,\frac{\pi }{2})$
So, $\sqrt{\sin 1}>\sin 1>0$ and $\sqrt{\cos 1}>\cos 1>0$
As $\sin x$ and $\cos x$ are positive and proper fractions in $x\in (0,\frac{\pi }{2})$

$\therefore \sqrt{\sin 1}+\sqrt{\cos 1}>\sin 1+\cos 1$
$\Rightarrow y>x...\left(i\right)$

Also,
$\sin 1+\cos 1>\sin 1-\cos 1(\because \cos 1>0)$
$\Rightarrow x>z...\left(ii\right)$

From $\left(i\right)$ and $\left(ii\right)$
$y>x>z$