Use linear approximation and differential to approximate: sin 31`

Question

Use linear approximation and differential to approximate: sin 31`

Answer 1

It saves some work to convert the 31° to radians   31º =  π/6 +  π/180    

 

sin( π/6 + π/180) ~  sin(π/6) +  cos(π/6) [π/180]      

        { because the derivative of sin is cos }

   The above is the first two terms of a Taylor series expansion of sin about the value π/6

    Since   sin(π/6) = 1/2  and cos(π/6) = .866  (√3 / 2 )

 

  sin(31º)  ~ 1/2 +  .866 π /180  ~   1/2 + .0151