Question

How do you convert ${{20}^{\circ }}$ into radians?

Answer 1

The foremost thing that we need to know to solve this problem is the definition of a radian. After doing so, we develop the relation between radians and degrees and then apply a unitary method to solve for ${{20}^{\circ }}$ .

Complete step by step answer:
The basic definition of one radian states that it is the angle subtended at the centre of the circle by an arc of length equal to its radius. As the length of the arc changes, the angle subtended changes. Larger the arc, greater is the angle subtended at the centre and vice-versa. Thus, the angle in the radian is measured by dividing the arc length by radius of the circle.
$Angle\text{ }in\text{ }radians=\dfrac{Length\text{ }of\text{ }arc}{Radius\text{ }of\text{ }the\text{ }circle}$
Then, the angle subtended by a semicircle at the centre will be
$\theta =\dfrac{\pi \times r}{r}$
= $\pi$ radians
But, we also know that the angle subtended at the centre of the circle by a semi-circular arc is ${{180}^{\circ }}$. This means,
${{180}^{\circ }}=\pi$ radians..... Equation 1
This is the standard relation between degrees and radians. By unitary method or, simply dividing each side of the equation 1 by 180, we get,
${{1}^{\circ }}=\dfrac{\pi }{180}$radians..... Equation 2
In the given problem, we need to find the value of ${{20}^{\circ }}$ in radians. Therefore, multiplying $20$ on each side of equation 2, we get,
${{20}^{\circ }}=\dfrac{\pi }{180}\times 20$radians
= $\dfrac{\pi }{9}$ radians
= $0.349$ radians
Therefore, we can conclude that ${{20}^{\circ }}=0.349$ radians.

Note:
Most of the students commit a mistake during the conversion of degrees to radians or vice-versa. There is often a confusion between $\dfrac{\pi }{180}$ and $\dfrac{180}{\pi }$. We need to take care of this.