Question

Which of the following is correct:
A. $\sin {1^\circ } > \sin {1^{}}$
B. $\sin {1^\circ } < \sin 1$
C. $\cos {1^\circ } < \sin 1$
D. None of these

Answer 1

Hint: To solve these questions, it is necessary to know the conversion of an angle from radians to degrees. After changing the angle of both the terms to one system, compare their values and then check which term is greater than the two.

Formula used: The following formulae can be used while solving these questions:
 $1$ Radian $= \dfrac{{{{180}^\circ }}}{\pi }$
 $1$ Degree $= \dfrac{\pi }{{{{180}^\circ }}}$

Complete step-by-step solution:
The relation between the radian angle system and the degree angle system is given as below,
$1$ Radian $= \dfrac{{{{180}^\circ }}}{\pi }$ or $1$ Degree $= \dfrac{\pi }{{{{180}^\circ }}}$
Now, in the term, $\sin 1$ the angle is $1$ radian whereas in the term $\sin {1^\circ }$ the
angle is in degrees.
From the above formula, we can calculate that
$1$ Radian $= \dfrac{{{{180}^\circ } \times 7}}{{22}}$
$= {57.2958^\circ }$
We know that the value of $\sin$ of an angle is always between $- 1$ and $1$ , and the value also increases as we increase the angle.
Considering the first option, $\sin {1^\circ } > \sin {1^{}}$. Substituting the value of $1$ Radian in the given option, we get,
$\sin {1^\circ } > \sin {57.298^\circ }$, therefore we reach a contradiction here to the point that the value of $\sin$ always increases as the angle increases. Hence, this option is incorrect.
Considering the second option, $\sin {1^\circ } < \sin 1$. Again, substituting the value of $1$ Radian in the given option, we get,
$\sin {1^\circ } < \sin {57.298^\circ }$, which is true, since the value of the $\sin$ ratio always increases as the angle increases. Hence, this option is correct.
Considering the third option, $\cos {1^\circ } < \sin 1$ . Again, substituting the value of $1$ Radian in the given option, we get,
$\cos {1^\circ } < \sin {57.298^\circ }$, which is again not true, since the value of the $\sin$ ratio always increases as the angle increases and the value of the $\cos$ ratio decreases with an increase in the angle. Hence, this option is incorrect.

Therefore, the correct answer will be option B. $\sin {1^\circ } < \sin 1$.

Note: To solve these questions, one must remember the relation between degrees and radians. Also, keep in mind to check the relation between the angles and their values for different trigonometric ratios.