What is the equation of the tangent line of #r=2cos(theta-pi/2) + sin(2theta-pi)# at #theta=pi/4#?

Answer 1

#rsin(theta-61.325^o)=-(sqrt2-1)sin(16.325^o)#. The Cartesian
form is #0.877x-0.480y-0.116=0#. See Socratic depiction.

Use #rsin(theta-psi)=asin(alpha-psi)#, for the equation to the
tangent at #P(a, alpha)#, where the slope of the tangent
#m=tanpsi=(r'sintheta+rcostheta)/(r'costheta-rsintheta)#.

Here,

#r = 2sintheta-sin2theta#
At #theta = pi/4=alpha, r = a =sqrt2-1#.
So, the point of contact P of the tangent is #(sqrt2-1, 45^o)#.
#r'=2costheta-2cos2theta=sqrt2#, at P.

The slope of the tangent at P is

#m=tanpsi=((sqrt2)(1/sqrt2)+(sqrt2-1)(1/sqrt2))/((sqrt2)(1/sqrt2)-(sqrt2-1)(1/sqrt2))#
#=2sqrt2-1#
#psi=arctan(2sqrt2-1)=61.325^o#.

So, the equation to the tangent is

#rsin(theta-61.325^o)=-(sqrt2-1)sin(16.325^o)#
Expanding and using #(x, y)=r(costheta, sintheta)#, the Cartesian

form is

#0.877x-0.480y-0.116=0# .

graph{(sqrt(x^2+y^2)(x^2+y^2-2y)+2xy)(0.877x-0.48y-0.116)((x-.293)^2+(y-.293)^2-.009)=0 [-5, 5, -2.5, 2.5]}

The Cartesian form is used for the Socratic graph.

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Answer 2

To find the equation of the tangent line of ( r = 2\cos(\theta - \frac{\pi}{2}) + \sin(2\theta - \pi) ) at ( \theta = \frac{\pi}{4} ), first, express the polar equation in terms of Cartesian coordinates. Then, find the derivative ( \frac{dy}{dx} ) with respect to ( \theta ). Finally, use the point-slope form of a line equation to find the equation of the tangent 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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