Parametric Curves and Tangent line equations

In summary, to find an equation of the tangent line to the curve with parametric equations x=tsint and y=tcost at the point (0,-π), you can solve for t by setting x=0 and y=-π and finding the appropriate value for t. This value can then be used to find the equation for the tangent line.
  • #1
ms. confused
91
0
Find an equation of the tangent line to the curve with parametric equations x=tsint, y=tcost at the point (0,-π).

went dy/dt / dx/dt --> cost - tsint/sint + tcost

t not given so figured it could be:

x=t(sin(1)) --> t= x/sin(1)

or

y=t(cos(1)) --> t= y/cos(1)

wondering if this is the right way to do it...looks really messy...help would be nice. Thanks.
 
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  • #2
ms. confused said:
t not given so figured it could be
You can find t by checking which t is necessary to be get at (0,-pi). If x has to be 0, either t or sint has to be zero. But with t = 0, y is 0 as well. So sint has to be zero, this is for t = 0 (but that couldn't be) or t = pi. Use t = pi and x will be 0 while y will be pi*cos(pi) = pi*(-1) = -pi, as was given. Then you have your t :smile:
 

1. What are parametric curves and how are they different from regular curves?

Parametric curves are curves that are defined by a set of equations, known as parametric equations, in terms of one or more parameters. These parameters are usually represented by the variables t and/or s. Unlike regular curves, which are defined by a single equation in terms of x and y, parametric curves can have more complex shapes and can be defined in higher dimensions.

2. How do you find the tangent line equation to a parametric curve at a given point?

To find the tangent line equation to a parametric curve at a specific point, you can use the parametric equations to find the derivatives of x and y with respect to t. Then, substitute the t-value of the given point into these derivatives to find the slope of the tangent line. Finally, use the point-slope form of a line to write the equation of the tangent line.

3. Can parametric curves only be defined in Cartesian coordinates?

No, parametric curves can also be defined in polar coordinates, cylindrical coordinates, and spherical coordinates. In these cases, the parametric equations will involve the parameters r, θ, ρ, and/or φ, depending on the coordinate system being used.

4. How are parametric curves used in real-world applications?

Parametric curves have many practical applications in fields such as physics, engineering, computer graphics, and animation. They are often used to model the motion of objects, such as projectiles or planets, and to create smooth and realistic animations in video games and movies.

5. Are there any disadvantages to using parametric curves?

One potential disadvantage of using parametric curves is that they can be more complex and difficult to work with compared to regular curves. Additionally, working with multiple parameters can make it challenging to visualize and manipulate the curve. However, parametric curves offer more flexibility and can accurately represent more complex shapes, making them a valuable tool in many applications.

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