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Question: Express the following angles in radian measure: (i) \(520^\circ \) (ii) \( - 310^\circ \) (i...

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

Explanation

Solution

We know the relation between the radian and degrees as 1∘=π180 radian1^\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. π180\dfrac{\pi }{{180}} . For changing the angle in minutes to radian, use the equation 1′=160∘=π180×160 radian1' = \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∘,−310∘,630∘ and −22∘30′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 360360 units called degrees. So, if the rotation from the initial side to the terminal side is 1360th\dfrac{1}{{360}}th of a revolution, then the angle is said to have a measure of one degree. It is denoted as 1∘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 2π2\pi
⇒2π=360∘⇒1∘=π180 radian\Rightarrow 2\pi = 360^\circ \Rightarrow 1^\circ = \dfrac{\pi }{{180}}{\text{ radian}}
For (i), we are given an angle of 520∘520^\circ
⇒520∘=π180×520=π9×26\Rightarrow 520^\circ = \dfrac{\pi }{{180}} \times 520 = \dfrac{\pi }{9} \times 26
Therefore, we get the angle as 520∘=26π9520^\circ = \dfrac{{26\pi }}{9}
For (ii), we are given an angle of −310∘- 310^\circ
⇒−310∘=π180×(−310)=π18×(−31)\Rightarrow - 310^\circ = \dfrac{\pi }{{180}} \times \left( { - 310} \right) = \dfrac{\pi }{{18}} \times \left( { - 31} \right)
Therefore, we get the angle as −310∘=−31π18 - 310^\circ = \dfrac{{ - 31\pi }}{{18}}
For (iii), we are given an angle of 630∘630^\circ
⇒630∘=π180×630=π2×7\Rightarrow 630^\circ = \dfrac{\pi }{{180}} \times 630 = \dfrac{\pi }{2} \times 7
Therefore, we get the angle as 630∘=7π2630^\circ = \dfrac{{7\pi }}{2}
For (iv), we are given an angle of −22∘30′ - 22^\circ 30'
Here 30′30' represents 3030 minutes. One degree angle is further divided into 6060 minutes
⇒1∘=60′⇒1′=160∘⇒30′=160×30=12∘\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:
⇒−22∘30′=π180×(−(22+12))=π180×(−452)=−π8\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 6060 minutes and one minute is equal to 6060 seconds. Similarly, while measuring angles, one degree is equal to 6060 minutes denoted as 1∘=60′1^\circ = 60'. And one minute is equal to 6060 seconds denoted as 1′=60′′1' = 60''. A negative angle simply implies that instead of going in an anti-clockwise direction, the measurement is done in a clockwise direction.