# Parametric equations for the tangent line

• Whatupdoc
In summary, parametric equations for the tangent line refer to a set of equations that use parameters to find the slope and intercept of a tangent line to a curve at a specific point. These equations are different from regular equations in that they use parameters instead of a single variable, making them more useful for dealing with complex curves. To find the slope of the tangent line, one can use the formula dy/dx = dy/dt / dx/dt. Parametric equations can be used for any type of curve and have real-world applications in fields such as physics, engineering, and computer graphics.
Whatupdoc
Find parametric equations for the tangent line at the point (cos(-4pi/6),sin(-4pi/6),-4pi/6) on the curve x=cost, y=sint,z=t

x(t) = _________
y(t) = _________
z(t) = _________

r'(t) = <-sin(t), cos(t), 1>
r'(0) = <0,1,1>

x = cos(-4pi/6) + 0t
y = sin(-4pi/6) +1t
z = -4pi/6 +t

the only part i got correct is the z, anyone know what I am doing wrong?

You want the derivative at the tangent point, not t=0.

yea your right, i don't know what i was thinking, thanks

## 1. What are parametric equations for the tangent line?

Parametric equations for the tangent line refer to a set of equations that can be used to find the slope and intercept of a tangent line to a curve at a specific point. These equations involve the use of parameters, typically denoted as t or u, to represent the independent variable.

## 2. How are parametric equations for the tangent line different from regular equations?

The main difference between parametric equations for the tangent line and regular equations is that parametric equations use parameters to represent the independent variable, while regular equations use a single variable, typically x or y. Parametric equations are also more useful when dealing with curves or functions that cannot be easily expressed as a single equation.

## 3. How do you find the slope of the tangent line using parametric equations?

To find the slope of the tangent line using parametric equations, you can use the formula: dy/dx = dy/dt / dx/dt. This means that the slope of the tangent line is equal to the derivative of the y-coordinate with respect to the parameter, divided by the derivative of the x-coordinate with respect to the parameter.

## 4. Can parametric equations be used for any type of curve?

Yes, parametric equations can be used for any type of curve, including circles, ellipses, parabolas, and more complex curves. This is because the use of parameters allows for a more flexible representation of the curve, compared to regular equations that may be limited in their ability to describe certain curves.

## 5. What are some real-world applications of parametric equations for the tangent line?

Parametric equations for the tangent line have various applications in fields such as physics, engineering, and computer graphics. Some examples include determining the trajectory of a projectile, finding the slope of a rollercoaster track at a particular point, and creating smooth curves in animation software.

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