Question

Convert $4$ radians into degree measure and also convert $-{{47}^{\circ }}37'$ into radian measure.

Answer 1

Hint: We know that radians and degree are the units of measuring angle. But why do we need different units for the same quantity? The answer is simple, for big angles the bigger unit is better suited and for small angles, smaller unit is suited. All we need to know is the conversion from one unit to the other.

Complete step-by-step answer:
When we are scaling, we are either scaling from bigger units to the smaller ones, or we are scaling smaller units to bigger ones. In other words, we are either zooming in or zooming out of the object under consideration.
This thing is used in our daily life, without us even noticing to this, like whenever we are travelling and using maps on our phones, we would like to zoom in and the look closely how the actual path is looking or whenever we are drawing a sketch, we sometimes stop and measure the comparable objects around and draw accordingly for more accuracy.
Over here, we need to convert $4$ radians to degree measure.
For this conversion, we need a scale to convert radians to degrees. So, by the definition of radians:
$\pi ={{180}^{\circ }}$ , where$\pi $ is in radians
$1rad={{\left( \dfrac{180}{\pi } \right)}^{\circ }}$ $-\left( 1 \right)$
Therefore, multiplying $4$ in the equation – (1), we get:
$4\left[ 1rad={{\left( \dfrac{180}{\pi } \right)}^{\circ }} \right]$
$\Rightarrow 4rad={{\left( \dfrac{720}{\pi } \right)}^{\circ }}$
Therefore, $4$ radians are equivalent to ${{\left( \dfrac{720}{\pi } \right)}^{\circ }}$ .
For the reverse calculation, we will follow from the equation $-\left( 1 \right)$ :
$\pi ={{180}^{\circ }}$
$\Rightarrow {{1}^{\circ }}=\dfrac{\pi }{180}rad$ $-\left( 2 \right)$
We know that $1$ degree is equivalent to $60$ minutes and therefore, $1$ minute is equivalent to $\dfrac{1}{60}$ degrees.
Hence, $37'={{\left( \dfrac{37}{60} \right)}^{\circ }}$
Therefore, multiplying equation $-\left( 2 \right)$ $-47\dfrac{37}{60}$ , we get:
${{\left( -47\dfrac{37}{60} \right)}^{\circ }}=\dfrac{\pi }{180}\times -47\dfrac{37}{60}$
$\Rightarrow -{{47}^{\circ }}37'=-\dfrac{2858}{10800}\pi \cdot rad$

Note: We must always remember the correct conversion scale in the conversion from one unit to the other. We must be very careful in writing the conversion scale, and not to confuse conversion scale with some other scale. Different units are used to measure different amounts of quantities. For length, two very different units are Kilometers and Miles.