Question

Find the degree measure corresponding to the following radian measures.
(a) $\dfrac{11}{16}$
(b) $-4$
(c) $\dfrac{5\pi }{3}$
(d) $\dfrac{7\pi }{6}$

Answer 1

Hint:The given problem is related to unit conversion. To convert from radian to degree we have to multiply by either $\dfrac{180}{\pi }$ or $\dfrac{180\times 7}{22}$ .

Complete step-by-step answer:
In the question, we are given certain radians and we have to convert them into their respective degree 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 the given radian to be converted are:
(i) $\dfrac{11}{16}$
So, to change it we have to multiply by $\dfrac{180}{\pi }$ or $\dfrac{180}{\dfrac{22}{7}}$ or $\dfrac{180\times 7}{22}$ so we get,
$\left( \dfrac{11}{16}\times \dfrac{180\times 7}{22} \right)$ $\Rightarrow \dfrac{315}{8}$
So, the value is equal to ${{\left( \dfrac{315}{8} \right)}^{{}^\circ }}$ .

(ii) $-4$
So, to change it we have to multiply by $\dfrac{180}{\pi }$ or $\dfrac{180}{\dfrac{22}{7}}$ or $\dfrac{180\times 7}{22}$ so we get,
$\left( -4\times \dfrac{180\times 7}{22} \right)$ $\Rightarrow \dfrac{-2520}{11}$
So, the value is equal to ${{\left( \dfrac{-2520}{11} \right)}^{{}^\circ }}$

(iii) $\dfrac{5\pi }{3}$
So, to change it we have to multiply by $\dfrac{180}{\pi }$ to convert the value of radian to degree.
$\dfrac{5\pi }{3}\times \dfrac{180}{\pi }$ $\Rightarrow 300$
So, the value is equal to $300{}^\circ$ .
(vi) $\dfrac{7\pi }{6}$
So, to change it we have to multiply by $\dfrac{180}{\pi }$ to convert the value of radian to degree.
$\dfrac{7\pi }{6}\times \dfrac{180}{\pi }$ $\Rightarrow 120$
So, the value is equal to $120{}^\circ$ .

Note: Students should think before multiplying by either $\dfrac{180}{\pi }$ or $\dfrac{180\times 7}{22}$ . The radians if are given in terms of $\pi$ should be multiplied by $\dfrac{180}{\pi }$ while rest by $\dfrac{180\times 7}{22}$ .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.