Question

Which of the following option(s) is correct?
(a) $ \sin {{1}^{\circ }} < \sin 1 $
(b) $ \sin {{1}^{\circ }} > \sin 1 $
(c) $ \sin {{1}^{\circ }}=\sin 1 $
(d) $ \sin {{2}^{\circ }} > \sin 2 $

Answer 1

Hint: We start solving the problem by recalling the conversion between degrees and radians as 1 rad = $ {{57.3}^{\circ }} $ . We then convert the given 1 rad and 2 rad in term of degrees using the conversion. We then make use of the fact that sine function is increasing in the interval $ \left[ 0,{{90}^{\circ }} \right] $ i.e., $ \sin a < \sin b $ if $ a < b $ and $ a,b\in \left[ 0,{{90}^{\circ }} \right] $ to get the relation between $ \sin {{1}^{\circ }} $ and $ \sin 1 $ . We then make use of the result $ \sin \left( {{180}^{\circ }}-\alpha \right)=\sin \alpha $ to find the value of $ \sin 2 $ which helps us to get the relation between $ \sin {{2}^{\circ }} $ and $ \sin 2 $ .

Complete step by step answer:
According to the problem, we need to find which of the given option(s) is correct.
Let us recall the conversion between degrees and radians. We know that 1 rad = $ {{57.3}^{\circ }} $ .
So, we get $ \sin 1=\sin {{57.3}^{\circ }} $ and $ \sin 2=\sin {{114.6}^{\circ }} $ .
We know that sine function is increasing in the interval $ \left[ 0,{{90}^{\circ }} \right] $ . So, we have $ \sin a < \sin b $ if $ a < b $ and $ a,b\in \left[ 0,{{90}^{\circ }} \right] $ .
Since $ {{1}^{\circ }} < {{57.3}^{\circ }} $ , we get $ \sin {{1}^{\circ }} < \sin {{57.3}^{\circ }}\Leftrightarrow \sin {{1}^{\circ }} < \sin 1 $ .
This tells us that option (a) is correct.
Now, let us check the relation between $ \sin {{2}^{\circ }} $ and $ \sin 2 $ .
We have $ \sin 2=\sin {{114.6}^{\circ }} $ .
 $ \Rightarrow \sin 2=\sin \left( {{180}^{\circ }}-{{65.4}^{\circ }} \right) $ .
We know that $ \sin \left( {{180}^{\circ }}-\alpha \right)=\sin \alpha $ .
 $ \Rightarrow \sin 2=\sin {{65.4}^{\circ }} $ .
Since $ {{2}^{\circ }} < {{65.4}^{\circ }} $ , we get $ \sin {{2}^{\circ }} < \sin {{65.4}^{\circ }}\Leftrightarrow \sin {{2}^{\circ }} < \sin 2 $ .
This tells us that option (d) is not correct.
 $ \, therefore, $ The correct option for the given problem is (a).

Note:
 We should know whenever we see the angle represented with a real number without showing the degree, then it should be considered as a radian. We can also make use of the fact $ 1rad=\dfrac{{{180}^{\circ }}}{\pi } $ to solve this problem. We can also solve this problem by plotting the sine curve in the interval $ \left[ 0,{{180}^{\circ }} \right] $ and comparing the given values. Similarly, we can expect problems to find the relation between $ \sin {{3}^{\circ }} $ and $ \sin 3 $ .