Question

Find the radian measures corresponding to the following degree measure:
(a) $25{}^\circ $
(b) \[-47{}^\circ 30'\]
(c) \[240{}^\circ \]
(d) \[520{}^\circ \]

Answer 1

Hint:The given problem is related to unit conversion. To convert from degree to radian one should multiply by $\dfrac{\pi }{180}$ to get the value in radians.

Complete step-by-step answer:
In the question, we are given certain degrees and we have to convert them into their respective radian measures.
Before proceeding we will first briefly say something about radian.
The radian is a S.I. unit for measuring angles and is the standard unit of angular measure used in areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees.
Radian describes the plain angle subtended by a circular arc as the length of arc divided by radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. The magnitude in radius of such a subtend angle is equal to the ratio of the arc length to the radius of circle; that is $\theta =\dfrac{s}{r}$ , where $\theta $ is the subtended angle in radius, s is arc length and r is radius.
Conversely, the length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radius that is $s=r\theta $ .
Now, in the question given degrees to be converted are:
(i) \[25{}^\circ \]
So, to change from degree to radian we will multiply by $\dfrac{\pi }{180}$ so, we get
$25\times \dfrac{\pi }{180}$ $\Rightarrow \dfrac{5\pi }{36}$ radians

(ii) \[-47{}^\circ 30'\]
We know that $1{}^\circ $ means 60’ so, 30’ will be ${{\left( \dfrac{1}{2} \right)}^{{}^\circ }}$
Then we can write \[-47{}^\circ 30'\] as
$-\left( 47+\dfrac{1}{2} \right)=-{{\left( \dfrac{95}{2} \right)}^{{}^\circ }}$
Now to change from degree to radian, we will multiply by $\dfrac{\pi }{180}$ so, we get
\[\dfrac{-95}{2}\times \dfrac{\pi }{180}\text{ }\Rightarrow \dfrac{19\pi }{72}\] radians

(iii) $240{}^\circ $
Now, by changing from degree to radian we will multiply by $\dfrac{\pi }{180}$ so we get,
\[240\times \dfrac{\pi }{180}\] \[\Rightarrow \dfrac{4\pi }{3}\] radians

(iv) $520{}^\circ $
Now to change from degree to radian, we will multiply by $\dfrac{\pi }{180}$ so we get,
$520\times \dfrac{\pi }{180}\text{ }\Rightarrow \dfrac{26\pi }{9}$ radians

Note: The radians are generally given in terms of $\pi $ . We can change into decimal by substituting $\pi $ as 3.14 or $\dfrac{22}{7}$ and get the answer in decimal form. Generally this is used to represent the arc length if the angle given is $\theta $ then the value in radian represents its arc length subtended by it.Students should remember to convert from degree to radian one should multiply by $\dfrac{\pi }{180}$ to get the value in radians and to convert from radian to degree one should multiply by $\dfrac{180 }{\pi}$ to get the value in degrees.