Question: Convert \[{{40}^{{}^\circ }}{{40}^{'}}\] into radian....

Question

Convert ${{40}^{{}^\circ }}{{40}^{'}}$ into radian.

Answer 1

Solution

The radian, denoted by the symbol $rad$ , is the SI unit for measuring angles and the standard unit of angular measurement used in many fields of mathematics.The unit was previously a SI supplementary unit (which was abolished in 1995), and the radian is now SI derived unit. The radian is defined in the SI as a dimensionless value, so its symbol is frequently omitted.

Complete step by step answer:
One radian is defined as the angle formed by the center of a circle intersecting an arc of length equal to the radius of the circle. In general, the magnitude of a subtended angle in radians equals the ratio of the arc length to the radius of the circle; that is, $\theta =\dfrac{s}{r}$ , where is the subtended angle in radians, $s$ is the arc length, and $r$ is the radius.

The length of the intercepted arc, on the other hand, is equal to the radius multiplied by the magnitude of the angle in radians; that is, $s=r\theta$ . one radian equals to $\dfrac{{{180}^{{}^\circ }}}{\pi }$ Thus, to convert radians to degrees, multiply by $\dfrac{{{180}^{{}^\circ }}}{\pi }$ .

The steps below demonstrate how to convert an angle in degrees to radians.
Step 1: Write the numerical value of the angle's measure in degrees.
Step 2: Now, multiply the numeral value from step $1$ by $\dfrac{\pi }{180}$ .
Step 3: Simplify the expression by canceling the numerical common factors.
Step 4: The angle measured in radians will be the result of the simplification.

In the above example,
${{40}^{{}^\circ }}{{40}^{'}}=40+\dfrac{4}{6}=\dfrac{240+4}{6}=\dfrac{244}{6}$
$\Rightarrow {{40}^{{}^\circ }}{{40}^{'}}=\frac{244}{6}\times \dfrac{\pi }{180}$
$\therefore {{40}^{{}^\circ }}{{40}^{'}}=0.7095$ $rad$ .

Thus, ${{40}^{{}^\circ }}{{40}^{'}}$$$$=0.7095$ $rad$ .

Note: Angles are universally measured in radians in calculus and most other branches of mathematics beyond practical geometry. This is due to the mathematical "naturalness" of radians, which leads to a more elegant formulation of several important results.