What is the equation of the normal line of #f(x)= cscx# at #x = pi/8#?

Answer 1
Note that #f(pi/8)=csc(pi/8)#, so the normal line will pass through the point #(pi/8,csc(pi/8))#.
The slope of the normal line will be the inverse reciprocal of the slope of the tangent line at #x=pi/8#, since the lines are perpendicular.
To find the slope of the tangent line there, find the derivative of #f# at #x=pi/8#.
The derivative of #csc(x)#, if you don't have it memorized, is easiest found using the chain rule on #csc(x)=1/sin(x)=(sin(x))^-1#. (Note this isn't inverse sine— rather sine to the negative first power.)
#f(x)=(sin(x))^-1#
#=>f'(x)=-(sin(x))^-2*cos(x)=-cos(x)/sin^2(x)#
So the slope of the tangent line is #-cos(pi/8)/sin^2(pi/8)# and the slope of the normal line is #sin^2(pi/8)/cos(pi/8)#.
The line passing through #(pi/8,csc(pi/8))# with slope #sin^2(pi/8)/cos(pi/8)# is:
#y-1/sin(pi/8)=sin^2(pi/8)/cos(pi/8)(x-pi/8)#

We could find these decimals, but we can also find the exact values.

I like to use the double angle formulas:

#cos(2x)=2cos^2(x)-1#

So:

#cos(pi/4)=2cos^2(pi/8)-1#
#cos^2(pi/8)=1/2(cos(pi/4)+1)=1/2(1/sqrt2+1)=(1+sqrt2)/(2sqrt2)=(2+sqrt2)/4#

Then:

#cos(pi/8)=sqrt(2+sqrt2)/2#
The same process can be done using #cos(2x)=1-2sin^2(x)# to find that
#sin^2(pi/8)=(2-sqrt2)/4#
#sin(pi/8)=sqrt(2-sqrt2)/2#

Then the normal line becomes:

#y-2/sqrt(2-sqrt2)=(2-sqrt2)/4(2/sqrt(2+sqrt2))(x-pi/8)#
#y-2/sqrt(2-sqrt2)=(2-sqrt2)/(2sqrt(2+sqrt2))(x-pi/8)#
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Answer 2

The equation of the normal line of f(x) = csc(x) at x = pi/8 is y = -cot(pi/8)(x - pi/8) + csc(pi/8).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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