Question

Which of the following 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: Here \[\sin 1\], mentions the angle in radians. Thus with the help of a circular sector, establish the relation between degree and radius. Thus find the value of \[\sin {{1}^{\circ }}\] and \[\sin 1\]. Compare them and find who is greatest with the same relations compare \[\sin {{2}^{\circ }}\] and \[\sin 2\].

Complete step-by-step answer:

First let us compare between \[\sin {{1}^{\circ }}\] and \[\sin 1\].
Hence, \[\sin 1\] means the sine of 1 radius. It can be represented by placing C in power of that angle, i.e. \[{{1}^{C}}\].
We can define one radian as the angle in degrees for which the radius of a circular sector is equal to its length. We told that,
radius = length of a circular sector.
We now that length of circular sector \[=\dfrac{\theta }{{{360}^{\circ }}}\times 2\pi r\]
Thus we can say that,
\[r=\dfrac{\theta }{{{360}^{\circ }}}\times 2\pi r\]
Thus cancelling like terms we get,
\[\theta =\dfrac{{{180}^{\circ }}}{\pi }\], which is 1 radian.
The approximate value of 1 radian is i.e. \[{{1}^{C}}={{57.3}^{\circ }}\]
We know that sine is an increasing function in the first quadrant.
\[\begin{align}
  & \therefore \sin {{1}^{\circ }}=0.017 \\
 & \sin {{1}^{C}}=0.841 \\
\end{align}\]
Hence from the above we can see the value of sine in degrees and radians, which is correct to three decimals. Here, \[\sin {{1}^{\circ }}\] is greater than \[\sin {{1}^{\circ }}\] i.e. \[\sin {{1}^{\circ }}<\sin 1\].
\[\therefore \] Thus from the options given we can say that, \[\sin {{1}^{\circ }}<\sin 1\].
We got, \[{{1}^{C}}={{57.3}^{\circ }}\]
\[\therefore {{2}^{C}}=2\times {{57.3}^{\circ }}={{114.6}^{\circ }}\]
Thus the value of \[\sin {{2}^{\circ }}=0.0348\]
\[\sin {{2}^{C}}=0.909\]
By comparing the options we can say that only \[\sin {{1}^{\circ }}<\sin 1\] is correct.
\[\therefore \] Option (a) is the correct answer.

Note: If we compare between \[\sin {{1}^{\circ }}\] and \[\sin {{2}^{\circ }}\] degree, we get that \[\sin {{2}^{\circ }}\] is greater than \[\sin {{1}^{\circ }}\]. \[\sin {{1}^{\circ }}<\sin {{2}^{\circ }}\] i.e. 0.017 < 0.0348, similarly comparing between \[\sin 1\] and \[\sin 2\] radius, we can say that \[\sin 2\] is greater than \[\sin 1\].
\[\therefore \sin 1<\sin 2\] i.e. 0.841 < 0.909