Question

Which of the following is least? (All angles have been measured in radians)
A. $\sin 3$
B. $\sin 2$
C. $\sin 1$
D. $\sin 7$

Answer 1

Hint: First we will convert the given angles in degree measure by using the formula $1\text{ Rad}\times \dfrac{180}{\pi }=57.2{}^\circ $. Then we will compare the values of angles to find the least among all.

Complete step by step answer:
We have been given the measure of angles of sine.
We have to find the least from the given angles.
Now, we have given the measure of angles in radians as $\sin 3$, $\sin 2$, $\sin 1$, $\sin 7$.
Let us first convert the given angles into degree measures by using the formula $1\text{ Rad}\times \dfrac{180}{\pi }=57.2{}^\circ $. Then we will get
$\begin{align}
  & \Rightarrow 2\text{ Rad}\times \dfrac{180}{\pi }=114.6{}^\circ \\
 & \Rightarrow 3\text{ Rad}\times \dfrac{180}{\pi }=171.9{}^\circ \\
 & \Rightarrow 7\text{ Rad}\times \dfrac{180}{\pi }=401.1{}^\circ \\
\end{align}$
Now, we know that the value of $\sin $ increases from $0\text{ to }\dfrac{\pi }{2}$ in the first quadrant. As the quadrant changes the angle also changes, so let us convert the obtained angles into acute angles by subtracting from the respective quadrant angles. Then we will get
\[\begin{align}
  & \Rightarrow \sin 1=57.3{}^\circ \\
 & \Rightarrow \sin 2=180{}^\circ -114.6{}^\circ =65.4{}^\circ \\
 & \Rightarrow \sin 3=180{}^\circ -171.9{}^\circ =8.1{}^\circ \\
 & \Rightarrow \sin 7=401.1{}^\circ -360{}^\circ =41.1{}^\circ \\
\end{align}\]
Now, when we compare the obtained values of angles we will get
 $\begin{align}
  & \Rightarrow 8.1{}^\circ < 41.1{}^\circ < 57.3{}^\circ < 65.4{}^\circ \\
 & \Rightarrow \sin 3 < \sin 7 < \sin 1 < \sin 2 \\
\end{align}$
Hence we get $\sin 3$ as least angle.

So, the correct answer is “Option A”.

Note: The point to be noted is that when the angle changes the quadrant, angle changes from acute to obtuse. So it is necessary to convert the angles into acute form for easy comparison. Highest value of angle has the highest sin value and least angle has least sine value.