Question

Convert 1 radian into a degree minute and second?

Answer 1

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 radian. We use the concept of the unitary system. We also use multiplication for the relation of ${{1}^{\circ }}$ being equal to 60 minutes and 1 minute $\left( {{1}^{'}} \right)$ is equal to 60 seconds $\left( {{1}^{''}} \right)$.

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 also know that ${{1}^{\circ }}$ is equal to 60 minutes and 1 minute $\left( {{1}^{'}} \right)$ is equal to 60 seconds $\left( {{1}^{''}} \right)$. 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 get that 1 radian is equal to ${{\left( \dfrac{180}{\pi } \right)}^{\circ }}$.
${{\left( \dfrac{180}{\pi } \right)}^{\circ }}={{\left( 57\dfrac{3}{11} \right)}^{\circ }}$. We now convert ${{\left( \dfrac{3}{11} \right)}^{\circ }}$ into minutes.
${{\left( \dfrac{3}{11} \right)}^{\circ }}={{\left( \dfrac{3}{11}\times 60 \right)}^{'}}={{\left( 16\dfrac{4}{11} \right)}^{'}}$
We now convert ${{\left( \dfrac{4}{11} \right)}^{'}}$ into seconds.
${{\left( \dfrac{4}{11} \right)}^{'}}={{\left( \dfrac{4}{11}\times 60 \right)}^{''}}={{\left( 21\dfrac{9}{11} \right)}^{''}}$
Therefore, 1 radian is equal to ${{57}^{\circ }}{{16}^{'}}{{\left( 21\dfrac{9}{11} \right)}^{''}}$.

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.