What is the equation of the tangent line of #r=5theta + 2sin(4theta+(2pi)/3) # at #theta=(2pi)/3#?

Answer 1

#rsin(theta^o-20.3^o)=8.74sin(99.7^o)#

I recall that, I have given here the formula, for the polar equation io

the tangent at #P( a, alpha )#, with slope
#m =tanpsi=(r'sin theta + r cos theta)/(r'costheta-rsintheta)#,
evaluated at #(a, alpha)#, as
#rsin(theta-psi)=asin(alpha-psi)#

Here,

#r =5theta+2sin(4theta+2/3pi)#
#alpha=2/3pi=2.0944 radian =120^o#
#a = 10/3pi -sqrt3=8.74#
#r' =5=8cos(4theta+2/3pi) =1,# at P
#m=0.3400 and psi=20.3^o#.
Now, the equation to the tangent at #P(8.74, 120^o)# is
#rsin(theta^o-20.3^o)=8.74sin(120^o-20.3^o)=8.74sin(99.7^o)#
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Answer 2

To find the equation of the tangent line of (r = 5\theta + 2\sin(4\theta + \frac{2\pi}{3})) at (\theta = \frac{2\pi}{3}), we'll first find the polar coordinates of the point of interest by substituting (\theta = \frac{2\pi}{3}) into the equation to find (r). Then, we'll differentiate the polar equation with respect to (\theta) to find the slope of the tangent line at that point. Finally, we'll use the point-slope form of the equation of a line to find the equation of the tangent line.

After finding the slope, the equation of the tangent line is:

[ r = -4\theta + 5 \sqrt{3} ]

Where ( r ) is in terms of ( \theta ).

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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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