algebraically | we know sin'=cos and cos(pi/3)=1/2 |
analytically | we know sin(x) = x-x^3/3! + x^5/5! - ... and so
sin'(x) = 1-x^2/2! + x^4/4!-... = cos(x) so that sin'(pi/3)=1/2. |
geometrically | sin(x) is the height of a right angle triangle with
hypotenuse of length 1. The rate of change of this
length in dependence of the angle can be seen geometrically. |
historically | the derivative can be derived from Euler's formula exp(i x) = cos(x) + i sin(x) which has the
derivative i exp(i x) = i cos(x) - sin(x). Comparing real
and complex parts shows cos'=-sin, sin'=cos. |
graphically | draw the graph of sin(x) and determine the slope at x=pi/3 |
numerically | take a small number 1/1000 and compute 1000 sin(pi/3+1/1000)-sin(pi/3))
which gives 0.499567. |
conceptually | since sin(x) increases for increasing for acute angles, the result is positive. |
psychologically | my teacher does not like to assign problems with irrational numbers as answers.
The result should be a simple rational number. Because 0 and 1 are out of question,
the next reasonable result is 1/2 ...... |
experimentally | here is an esoteric experiment: why not use Fourier series and
differentiate that series. To make it interesting, take f(x) = |sin(x)|
since sin(x)=|sin(x)| around pi/3 ... |