Question

Express the given angle ${{45}^{\circ }}20' 10''$ in rad measure $\left( \pi =3.1415 \right)$.

Answer 1

In the question, try to convert the whole measure of angle in degree by using the fact as 1 minute or $1'$ is equal to 60 seconds or $60’’$ and 1 degree or ${{1}^{\circ }}$ is equal to 60 minutes or $60’$ . Then change the given ${{45}^{\circ }}20' 10''$ to ${{\left( \dfrac{16321}{360} \right)}^{\circ}}$ . Then change it to radian by multiplying it by $\dfrac{\pi }{180}$ .

Complete step-by-step solution:
In the question, we are given an angle, which is measured ${{45}^{\circ }}20' 10''$ and we have to express it in the form of a radian.
Before proceeding we will first briefly say something about radian.
The radian is an 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.
The unit is formerly an SI supplementary unit, but this category was abolished in 1995 and the radian is now considered as an SI derived unit.
Radian describes the plane angle subtended by a circular arc as the length of arc divided by the 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 magnitude in radians of such a subtend angle is equal to the ratio of the arc length to the radius of a circle; that is $\theta \ =\ \dfrac{s}{r}$, where $\theta$ is the subtended angle in radians, s is arc length and r is the radius.
First, we will convert it into degrees. So, we know that an angle is given as ${{45}^{\circ }}20' 10''$.
We know that $1' =60''$. So, we can write it as $60'' =1'$. Then we can say that $10'' =\left( \dfrac{1}{6} \right)'$.
So, the given degree will be now,
${{45}^{\circ }}\left( 20+\dfrac{1}{6} \right)'$ or ${{45}^{\circ }}\left( \dfrac{121}{6} \right)'$.
Now we know that ${{1}^{\circ }}=60'$. So, we can say write it as $60' ={{1}^{\circ }}$. Then we can say that $1' ={{\left( \dfrac{1}{60} \right)}^{\circ }}$.
Hence $\left( \dfrac{121}{6} \right)' ={{\left( \dfrac{121}{6}\times \dfrac{1}{60} \right)}^{\circ }}$, which is equal to ${{\left( \dfrac{121}{360} \right)}^{\circ }}$.
So, the total degree is equal to ${{45}^{\circ }}+{{\dfrac{121}{360}}^{\circ }}$ which is equal to $\dfrac{45\times 360+121}{360}$ or $\dfrac{16321}{360}$.
Now, we have to change ${{\left( \dfrac{16321}{360} \right)}^{\circ }}$ into radian so we will do it by multiplying by $\dfrac{\pi }{180}$ .
So, we get,
$\dfrac{16321}{360}\times \dfrac{\pi }{180}$
Hence, on calculating we get $\dfrac{16321\pi }{64800}$
So, the answer is $\dfrac{16321\pi }{64800}$.

Note: Students generally make mistakes while changing seconds to minutes and minutes to degree and also confuse between signs too.
We can also solve this problem further by putting the value of $\pi$ in it,
$\dfrac{16321\pi}{64800}=\dfrac{16321\times3.1415}{64800}=\dfrac{56331.9315}{64800}=0.86931$
And also keep in mind when you convert a degree into radian we need to multiply it by $\dfrac{\pi }{180}$.