SOLUTION: Find the equation of the normal line to y=6.5sin(2.5x) at x=pi/4 radians

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Question 889292: Find the equation of the normal line to y=6.5sin(2.5x) at x=pi/4 radians
Found 2 solutions by Fombitz, Edwin McCravy:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Take the derivative of the function and find its value at x=pi%2F4.
This value is the slope of the tangent line which is perpendicular to the normal line.
y=6.5sin%282.5x%29
dy%2Fdx=2.5%286.5%29cos%282.5x%29
dy%2Fdx%28pi%2F4%29=%2865%2F4%29cos%28%285pi%29%2F8%29
dy%2Fdx%28pi%2F4%29=%2865%2F4%29%28-0.3827%29
m%5Bt%5D=-6.2186
Perpendicular line slopes are negative reciprocals.
m%5Bn%5D%2Am%5Bt%5D=-1
m%5Bn%5D=-1%2F%28-6.2186%29
m%5Bn%5D=0.161
Use the point slope form of a line,
y-6.005=0.161%28x-0.7854%29
y-6.005=0.161x-0.1264
y=0.161x%2B5.879
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Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
First we find the slope of the green tangent line by taking the
derivative of the equation

y%22%22=%22%226.5sin%282.5x%29%29

%28dy%29%2F%28dx%29%22%22=%22%226.5cos%282.5x%29%2A2.5

%28dy%29%2F%28dx%29%22%22=%22%2216.25cos%282.5x%29

We substitute x=pi%2F4

matrix%281%2C3%2C%28dy%29%2F%28dx%29%2Cat%2Cpi%2F4%29%22%22=%22%2216.25cos%282.5%2Aexpr%28pi%2F4%29%29%22%22=%22%22-6.218599

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%22%22=%22%220.1608079247

The point is %28matrix%281%2C3%2Cpi%2F4%2C%22%2C%22%2C6.5sin%286.5%2Aexpr%28pi%2F4%29%29%29%29%22%22=%22%22%28matrix%281%2C3%2C0.7853981634%2C%22%2C%22%2C6.005216961%29%29      

Using the point-slope formula:

y-y%5B1%5D%22%22=%22%22m%28x-x%5B1%5D%29

y-6.005216961%22%22=%22%220.1608079247%28x-0.7853981634%29

That simplifies to

y%22%22=%22%220.1608079247x+%2B+5.878918713

Edwin