What is the equation of the tangent line to the polar curve # f(theta)=theta^2cos(3theta)-thetasin(2theta)+tan(theta/6) # at #theta = pi#?

Answer 1

#Y = ((pi^2-1/sqrt[3]) (1/sqrt[3] - pi^2 + X))/( 4 pi-2/9)#

The pass equations are

#{ (x(theta) = r(theta)cos(theta)), (y(theta)=r(theta)sin(theta)) :}#

The tangent space is obtained making

#(dy)/(dx) = ((dy)/(d theta))((d theta)/(dx))#

but

#(dx)/(d theta) = (dr)/(d theta) cos(theta) - r(theta)sin(theta)#
#(dy)/(d theta) = (dr)/(d theta) sin(theta) + r(theta)cos(theta)#

with

#r(theta) = theta^2 Cos(3 theta) - theta Sin(2 theta) + Tan(theta/6)#

Deriving and calculating for #theta = pi# we obtain

#((dy)/(dx))_{theta=pi} = ( pi^2-1/(sqrt[3]) )/( 4 pi-2/9 )#

#x_0 = x(pi) = -(1/sqrt[3]) + pi^2#
#y_0 = y(pi) = 0#

so the tangent line at #p_0 = {x_0,y_0}# is

#Y = ((pi^2-1/sqrt[3]) (1/sqrt[3] - pi^2 + X))/( 4 pi-2/9)#

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Answer 2

To find the equation of the tangent line to the polar curve ( f(\theta) = \theta^2 \cos(3\theta) - \theta \sin(2\theta) + \tan(\frac{\theta}{6}) ) at ( \theta = \pi ), follow these steps:

  1. Calculate the value of ( f(\theta) ) at ( \theta = \pi ) to find the point of tangency.
  2. Find the derivative of ( f(\theta) ) with respect to ( \theta ).
  3. Evaluate the derivative at ( \theta = \pi ) to find the slope of the tangent line.
  4. Use the point of tangency and the slope to write the equation of the tangent line in polar form.

Step 1: ( f(\pi) = \pi^2 \cos(3\pi) - \pi \sin(2\pi) + \tan(\frac{\pi}{6}) )

Step 2: Find the derivative of ( f(\theta) ): [ f'(\theta) = (2\theta \cos(3\theta) - 3\theta^2 \sin(3\theta)) - (\sin(2\theta) - 2\theta \cos(2\theta)) + \frac{1}{6\cos^2(\theta/6)} ]

Step 3: Evaluate the derivative at ( \theta = \pi ): [ f'(\pi) = (2\pi \cos(3\pi) - 3\pi^2 \sin(3\pi)) - (\sin(2\pi) - 2\pi \cos(2\pi)) + \frac{1}{6\cos^2(\pi/6)} ]

Step 4: Use the point of tangency ( (\pi, f(\pi)) ) and the slope ( f'(\pi) ) to write the equation of the tangent line in polar form.

So, the equation of the tangent line is:

[ r = \frac{f(\pi)}{\cos(\theta - \pi)} ]

where ( r ) is the distance from the origin, and ( \theta ) is the angle.

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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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