Question

How‌ ‌do‌ ‌you‌ ‌compare‌ ‌$\sin‌ ‌{{1}^{\circ‌ ‌}}$‌ ‌and‌ ‌$\sin‌ ‌1$?‌

Answer 1

Hint: We describe the relation between the degree and radians, two ways to express the angles. We find the relation that 180 degrees is equal to $\pi $ radian. We express the degree value of ${{1}^{\circ }}$ in radian. We find the comparison between $\sin {{1}^{\circ }}$ and $\sin 1$.

Complete step by step answer:
We need to find the relations between the degree and radians. There are two ways to express the angles. They are degrees and radians. The way to differentiate them is using the degree sign on the angle value.
If the angle is $x$, then it means it’s $x$ radian and if it’s given ${{x}^{\circ }}$, then that means $x$ degree.
The relation between these two units is that 180 degrees is equal to $\pi $ radian. The value of $\pi $ is the usual value where $\pi =3.14$. (approx.)
Therefore, $\pi \text{ rad}={{180}^{\circ }}$.
We can convert it into radian using the relation where 1 degree is equal to $\dfrac{\pi }{180}$ radian. This gives $x$ degree is equal to $\dfrac{\pi x}{180}$ radian.
We have been given to compare $\sin {{1}^{\circ }}$ and $\sin 1$.
We get that ${{1}^{\circ }}$ is equal to $\dfrac{\pi }{180}$ radian. Also $\sin 1$ indicates the radian value for the sin ratio.
We find the value for $\dfrac{\pi }{180}$ which is equal to $0.017\text{ rad}$.
In the interval of $\left( 0,\dfrac{\pi }{2} \right)$, the sin ratio is increasing which means the ratio value increases with the rise in the angle.
We get that $1>0.017$ which gives $\sin 1>\sin 0.017$.
Changing the value, we get $\sin 1>\sin {{1}^{\circ }}$.

Note: Degrees and radians are ways of measuring angles. A radian is equal to the amount an angle would have to be open to capture an arc of the circle's circumference of equal length to the circle's radius.