Q.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) $\sin 1^\circ = \dfrac{\pi}{18^\circ}\sin 1$

The core idea here is understanding the distinction between angle units (degrees vs. radians) and the behavior of the sine function in the first quadrant.

When an angle is written without a unit, it is conventionally understood to be in radians. Thus, we are comparing $\sin 1^\circ$ with $\sin(1 \text{ radian})$.

1. Convert $1$ radian into degrees for a direct comparison. We know that $\pi$ radians $= 180^\circ$.
   Important

   $1 \text{ radian} = \frac{180^\circ}{\pi}$

2. Using the approximation $\pi \approx 3.14159$, we find $1 \text{ radian} \approx \frac{180^\circ}{3.14159} \approx 57.295^\circ$.
3. Now we are comparing $\sin 1^\circ$ and $\sin 57.295^\circ$. …

When we encounter trigonometric functions like $\sin x$, the argument $x$ can be expressed in either degrees or radians. If no unit is specified, it is universally understood to be in radians. This distinction is crucial for comparing values.

The core idea here is to compare the magnitudes of the angles $1^\circ$ and $1$ radian, and then use the property of the sine function in the relevant quadrant. Both $1^\circ$ and $1$ radian fall within the first quadrant ($0^\circ$ to $90^\circ$ or $0$ to $\pi/2$ radians). In this quadrant, the sine function is strictly increasing. This means that if we have two angles, say $\alpha$ and $\beta$, such that $0 < \alpha < \beta < \pi/2$, then $\sin \alpha < \sin \beta$.

Our task simplifies to determining which angle, $1^\circ$ or $1$ radian, is larger.

1. Understand the Units:

   The notation $\sin 1^\circ$ clearly indicates that the angle is $1$ degree.

   The notation $\sin 1$ (without a degree symbol) implies that the angle is $1$ radian. This is a standard convention in mathematics, especially in calculus and higher-level trigonometry.

   Watch out

   A common mistake is to assume that $1$ in $\sin 1$ also means $1$ degree. Always remember that an angle without a unit symbol is in radians.

2. Convert to a Common Unit:

   To compare $1^\circ$ and $1$ radian, we need to express them in the same unit. Let's convert $1$ radian into degrees.

   We know the conversion factor: $\pi$ radians $= 180^\circ$.

   Therefore, $1$ radian $= \frac{180}{\pi}$ degrees.

3. Approximate the Value of $1$ Radian in Degrees:

   We use the approximation $\pi \approx 3.14159$.

   $1 \text{ radian} \approx \frac{180}{3.14159} \text{ degrees} \approx 57.2958^\circ$.

4. Compare the Angles: …