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

Hint: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.