Question 889292
{{{drawing(2000/7,800,-1,4,-7,7,graph(2000/7,800,-1,4,-7,7,6.5sin(2.5x)),

circle(0.78539816,6.00521696,0.15),circle(0.78539816,6.00521696,0.13),circle(0.78539816,6.00521696,0.11),circle(0.78539816,6.00521696,0.09),circle(0.78539816,6.00521696,0.07),circle(0.78539816,6.00521696,0.05),circle(0.78539816,6.00521696,0.03),circle(0.78539816,6.00521696,0.01),
green(line(-1,17.10789219,4,-13.98510281)),
red(line(-4,5.235687014,5,6.682958336))




)}}}<pre>
First we find the slope of the green tangent line by taking the
derivative of the equation

{{{y}}}{{{""=""}}}{{{6.5sin(2.5x))}}}

{{{(dy)/(dx)}}}{{{""=""}}}{{{6.5cos(2.5x)*2.5}}}

{{{(dy)/(dx)}}}{{{""=""}}}{{{16.25cos(2.5x)}}}

We substitute {{{x=pi/4}}}

{{{matrix(1,3,(dy)/(dx),at,pi/4)}}}{{{""=""}}}{{{16.25cos(2.5*expr(pi/4))}}}{{{""=""}}}{{{-6.218599}}}

That's the slope of the green tangent line.

We want the slope of the red normal line, which is the
negative reciprocal of {{{-6.218599}}} or 

{{{m}}}{{{""=""}}}{{{0.1608079247}}}

The point is {{{(matrix(1,3,pi/4,",",6.5sin(6.5*expr(pi/4))))}}}{{{""=""}}}{{{(matrix(1,3,0.7853981634,",",6.005216961))}}}      

Using the point-slope formula:

{{{y-y[1]}}}{{{""=""}}}{{{m(x-x[1])}}}

{{{y-6.005216961}}}{{{""=""}}}{{{0.1608079247(x-0.7853981634)}}}

That simplifies to

{{{y}}}{{{""=""}}}{{{0.1608079247x + 5.878918713}}}

Edwin</pre>