Equation of tangent line (rec. form) to a polar curve

In summary, to find the rectangular form of the equation of the tangent line to a polar curve, you need to first calculate the slope using the formula for dy/dx in polar form. Then, convert the polar coordinates of the point of tangency to rectangular coordinates. Finally, substitute the slope and rectangular coordinates into the equation for a line, y-y1=m(x-x1), to get the equation of the tangent line in rectangular form.
  • #1
System
42
0
Equation of tangent line (rec. form) to a polar curve!

Homework Statement



Quesiton:

Find the rectangular form of the equation of the tangent line to the polar curve r=cos^3(theta) at the point corresponding to theta=pi/4


Homework Equations





The Attempt at a Solution



How to do that?

I mean finding it in RECTANGULAR FORM !


i know that
[tex]\frac{dy}{dx}=\frac{\frac{dr}{d\theta} sin(\theta)+rcos(\theta)}{\frac{dr}{d\theta} cos(\theta) - r sin(\theta)}[/tex]

I will calculate dy/dx at theta=pi/4 , and this is easy ..

The problem here is that, how can I find the equation of the tangent line in RECATNGULAR FORM ??

The equation of the tangent line is :

y-y1=m(x-x1)

m = the slope , and this one will be calculated by using the formula ..

but what about x1 and y1?
 
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  • #2
Hi System! :smile:

(have a pi: π and a theta: θ and try using the X2 and X2 tags just above the Reply box :wink:)
System said:
y-y1=m(x-x1)

m = the slope , and this one will be calculated by using the formula ..

but what about x1 and y1?

(x1,y1) will be (r,θ) at θ = π/4. :wink:
 
  • #3


I do not think so.
so y=pi/4 ?! :S
 
  • #4
(what happened to that π i gave you? :confused:)

No, you have to convert (r,θ) to (x,y) at θ = π/4.
 
  • #5


ohhh i see now
x = r cosθ = cos^4 θ
y = r sinθ = cos^3θ sinθ

I will evaluate them at θ=pi/4 to get x1 & y1
and I have m from the formula of dy/dx in polar

I will substitute x1,y1 & m in the line equation and I will be finish, right?
 
  • #6
(just got up :zzz: …)

Right! :biggrin:
 

1. What is an equation of tangent line in rectangular form to a polar curve?

An equation of tangent line in rectangular form to a polar curve is a mathematical expression that represents a line that touches the polar curve at a specific point, known as the point of tangency. It is written in the form y = mx + b, where m is the slope of the tangent line and b is the y-intercept.

2. How do you find the slope of the tangent line to a polar curve?

The slope of the tangent line to a polar curve can be found by taking the derivative of the polar curve with respect to the angle theta. This can be done using the polar form of the chain rule, which states that dy/dx = (dy/dtheta) / (dx/dtheta).

3. Can the equation of a tangent line to a polar curve be written in terms of the angle theta?

Yes, the equation of a tangent line to a polar curve can be written in terms of the angle theta. This is because the slope of the tangent line and the coordinates of the point of tangency are dependent on the angle theta. However, it can also be converted to rectangular form if desired.

4. What is the point of tangency on a polar curve?

The point of tangency on a polar curve is the point where the tangent line touches the curve. This point has the same coordinates in both polar and rectangular form, and its position on the curve is determined by the angle theta.

5. How do you use the equation of a tangent line to approximate the coordinates of a point on a polar curve?

If the equation of a tangent line to a polar curve is known, the coordinates of a point on the curve can be approximated by substituting the value of theta into the equation. This will give the value of y at that point, which can then be converted to rectangular form to find the x-coordinate. This method is useful for estimating points on a polar curve that are difficult to find exactly.

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