What is the equation of the line that is normal to the polar curve #f(theta)=sin(2theta+pi) -theta# at #theta = pi/4#?

Answer 1

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

Find #r# when #theta=pi/4# #r=sin(2(pi/4)+pi) -theta=(-4-pi)/4#
Find #(dr) /(d theta):# #r=sin(2 theta + pi) -theta# #(dr) /(d theta)=cos(2 theta + pi) (2)-1=2cos(2 theta + pi) -1#
Substitute #theta=pi/4#: #=2cos(2(pi/4)+pi)-1=-1#
Find the derivative: #dy/(dx)=((-1)sin (pi/4) +((-4-pi)/4)cos (pi/4))/((-1)cos(pi/4) -((-4-pi)/4) sin (pi/4))=-8/pi-1#
Find the respective #x# and #y#. #x=rcos(theta)=((-4-pi)/4) (sqrt(2)/2)=(-4sqrt(2)-pi sqrt(2))/8#
#y=rsin(theta)=((-4-pi)/4) (sqrt(2)/2)=(-4sqrt(2)-pi sqrt(2))/8#
The slope of a normal line is the negative reciprocal of the derivative. #-1/(-8/pi-1)=pi/(8+pi)#
Putting it all together: #(y-y_1)=m(x-x_1)# #(y-(-4sqrt(2)-pi sqrt(2))/8)=(pi/(8+pi))(x-(-4sqrt(2)-pi sqrt(2))/8)#
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Answer 2

To find the equation of the line normal to the polar curve (f(\theta) = \sin(2\theta + \pi) - \theta) at (\theta = \frac{\pi}{4}), we first need to find the slope of the tangent to the curve at that point.

Differentiate (f(\theta)) with respect to (\theta) to find (f'(\theta)). Then, evaluate (f'(\frac{\pi}{4})) to get the slope of the tangent line.

The slope of the normal line will be the negative reciprocal of the slope of the tangent line.

Once you have the slope of the normal line and the point ((\theta, f(\theta))) where it passes through, you can use the point-slope form of the equation of a line to find the equation of the normal line.

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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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