Question

Express the following angles in radian measure:
(i) $520^\circ $
(ii) $ - 310^\circ $
(iii) $630^\circ $
(iv) $ - 22^\circ 30'$

Answer 1

Hint: We know the relation between the radian and degrees as $1^\circ = \dfrac{\pi }{{180}}{\text{ radian}}$ . Use this equation and the unitary method to convert the given angles in degree by multiplying the factor to the RHS, i.e. $\dfrac{\pi }{{180}}$ . For changing the angle in minutes to radian, use the equation $1' = \dfrac{1}{{60}}^\circ = \dfrac{\pi }{{180}} \times \dfrac{1}{{60}}{\text{ radian}}$

Complete step-by-step answer:
Here in the problem, we are given with four measures of angles $520^\circ , - 310^\circ ,630^\circ {\text{ and }} - 22^\circ 30'$ . They are all measured in degrees and we need to convert these angles in radian measure.
Before starting with a solution, we need to understand a few concepts related to measures of angles. A complete revolution, i.e. when the initial and terminal sides are in the same position after rotating clockwise or anticlockwise, is divided into $360$ units called degrees. So, if the rotation from the initial side to the terminal side is $\dfrac{1}{{360}}th$ of a revolution, then the angle is said to have a measure of one degree. It is denoted as $1^\circ $ .
When the measurement is done in radians, it got a little complicated. If we have a circle with a radius of one unit and mark an arc length of one unit, then the angle subtended by this one unit length arc at the center is of one radian. This way a full circle of the unit radius will have an angle of $2\pi $
$ \Rightarrow 2\pi = 360^\circ \Rightarrow 1^\circ = \dfrac{\pi }{{180}}{\text{ radian}}$
For (i), we are given an angle of $520^\circ $
$ \Rightarrow 520^\circ = \dfrac{\pi }{{180}} \times 520 = \dfrac{\pi }{9} \times 26$
Therefore, we get the angle as $520^\circ = \dfrac{{26\pi }}{9}$
For (ii), we are given an angle of $ - 310^\circ $
$ \Rightarrow - 310^\circ = \dfrac{\pi }{{180}} \times \left( { - 310} \right) = \dfrac{\pi }{{18}} \times \left( { - 31} \right)$
Therefore, we get the angle as $ - 310^\circ = \dfrac{{ - 31\pi }}{{18}}$
For (iii), we are given an angle of $630^\circ $
$ \Rightarrow 630^\circ = \dfrac{\pi }{{180}} \times 630 = \dfrac{\pi }{2} \times 7$
Therefore, we get the angle as $630^\circ = \dfrac{{7\pi }}{2}$
For (iv), we are given an angle of $ - 22^\circ 30'$
Here $30'$ represents $30$ minutes. One degree angle is further divided into $60$ minutes
$ \Rightarrow 1^\circ = 60' \Rightarrow 1' = \dfrac{1}{{60}}^\circ \Rightarrow 30' = \dfrac{1}{{60}} \times 30 = \dfrac{1}{2}^\circ $
So the given angle will become:
$ \Rightarrow - 22^\circ 30' = \dfrac{\pi }{{180}} \times \left( { - \left( {22 + \dfrac{1}{2}} \right)} \right) = \dfrac{\pi }{{180}} \times \left( { - \dfrac{{45}}{2}} \right) = - \dfrac{\pi }{8}$

Note: We measure time in hours, minutes, and seconds, where one hour is equal to $60$ minutes and one minute is equal to $60$ seconds. Similarly, while measuring angles, one degree is equal to $60$ minutes denoted as $1^\circ = 60'$. And one minute is equal to $60$ seconds denoted as $1' = 60''$. A negative angle simply implies that instead of going in an anti-clockwise direction, the measurement is done in a clockwise direction.