Transforming of degree measure to radian one and back

1.   To find a radian measure of any angle by its given degree measure it is necessary to multiply:
       a number of degrees by π / 180 ≈ 0.017453, a number of minutes – by π / (180 · 60 ) ≈0.000291,  
       a number of seconds  –  by π / (180 · 60 · 60 ) ≈ 0.000005 and to add the found products.

       E x a m p l e .  Find a radian measure of an angle 12° 30’ with an of the fourth accuracy
                               decimal place.

       S o l u t i o n .  Multiply  12  by  π / 180 : 12 · 0.017453≈ 0.2094. 
                               Multiply 30 by π / (180 · 60 ) : 30 · 0.000291 ≈ 0.0087. 
                               Now we find:
                               12°30’ ≈ 0.2094 + 0.0087 = 0.2181 rad.

2.   To find a degree measure of any angle by its given radian measure it is necessary to multiply
      a number of radians by 180° / π ≈ 57°.296 = 57°17’45” ( a relative error of the result will be ~ 0.0004%,
      that corresponds to an absolute error ~ 5” for a round angle 360° ).

      E x a m p l e .  Find a degree measure of an angle 1.4 rad. with an accuracy up to 1’.

      S o l u t i o n .  We’ll find consequently: 
                              1 rad ≈ 57°17’45” ;
                              0.4 rad ≈0.4 · 57°.296 = 22°.9184;
                              0°.9184 · 60 ≈ 55’.104;
                              0’.104 · 60 ≈ 6”. 
                              So,  0.4 rad ≈ 22°55’6” and hence:

                                                     1 rad ≈ 57°17’45”
                                              +                          
                                                   0.4 rad ≈ 22°55’6”
                                           _____________________

                                                  1.4 rad ≈ 80°12’51”

                              After rounding this result according to the required accuracy up to 1’
                              we have finally: 1.4 rad ≈ 80°13’.