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Find the equation of the normal to y = 1/sin(2x) at the point where x = pi/4

Guest Aug 11, 2017
 #1
avatar+7024 
0

\(x=\dfrac{\pi}{4}\\ \therefore y = \dfrac{1}{\sin\left(2\cdot\dfrac{\pi}{4}\right)}=1\\ \dfrac{dy}{dx}=(\csc(2x))'=-2\csc(2x)\cot(2x)\\ \dfrac{dy}{dx}|_{x=\frac{\pi}{4}}=-2(1)(0)=0\\ y-1=0(x-\dfrac{\pi}{4})\\ y=1\\ \therefore\text{The equation of the normal to }y=\dfrac{1}{\sin(2x)}\text{ when }x=\pi/4\text{ is }\boxed{y=1}\)

I think this is the first time I do an application question :P

MaxWong  Aug 11, 2017
 #2
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+1

I got the same answer!

 

Book says x = pi/4

 

*facepalm*

Guest Aug 11, 2017
 #3
avatar+7024 
0

Well, even textbooks have mistakes... :P

MaxWong  Aug 11, 2017
 #4
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0

Cambridge 0606

 

Their answers are out of this world sometimes. Can't wait for IB.

Guest Aug 11, 2017
 #5
avatar+90088 
+2

 

Find the equation of the normal to y = 1/sin(2x) at the point where x = pi/4

 

We can write this as

 

y = [sin(2x) ]-1 

 

When x  = pi/4, y =  1 / sin (2 (pi/4))  =  1 / sin (pi/2)  = 1

 

y '  =   -2 [ sin (2x)]-2 * [ cos(2x)]  =   [-2 cos (2x)] / [ sin (2x)] 2  =   -2cot (2x)csc(2x)

 

 

At pi/4....the slope of the tangent line  =   -2 [ cot (2(pi/4) ) csc (2(pi/4) )=

-2 [ cot (pi/2)] [csc(pi/2)]  =  0

 

So the equation of the tangent line  is   ...    y - 1  = 0 ( x - pi/4)  →  y  = 1

 

So...the  slope of a line normal to this will be undefined.....and the equation of the line will be  x  = pi/4

 

Here's a graph to show this : https://www.desmos.com/calculator/qyxqzuupbs

 

 

cool cool cool

CPhill  Aug 11, 2017
 #6
avatar+7024 
+2

oh lol I am careless... I did it like I am doing a tangent line question :P

MaxWong  Aug 12, 2017

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