Parametric Equations for Tangent Line at (cos 0pi/6, sin 0pi/6, 0pi/6)

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  • #1
weckod
13
0
need parametric equations to the tangent line at the point
(cos 0pi/6, sin 0pi/6, 0pi/6) on the curve x = cost, y = sint, z = t

x(t) = ?
y(t)=?
z(t)=?

now from my understanding, i have to find the derivatives of x, y, and z right? and i did this... now alll i should do is plug in the x, y, z pts? and get the answers? i don't know if the 0pi/6 is correct because it was printed in w/ the problem... it could be pi/6 only and not 0pi/6... could someone help maybe i did some kind of calculation error... if possible explain thanks
 
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  • #2
anyone know how to help here?
 
  • #3
Think of your function as a basic space curve:
[tex] \vec r\left( t \right) = \left\langle {\cos t,\sin t,t} \right\rangle [/tex]

where
[tex] \vec r\,'\left( t \right) = \left\langle { - \sin t,\cos t,1} \right\rangle [/tex]

Your point, [tex] \left( {1,0,0} \right) [/tex], can be represented by the positional vector [tex] \vec r \left( 0 \right) [/tex].

As you remember from earlier calculus, the tangent line will thus be parallel to
[tex] \vec r \, ' \left( 0 \right) = \left\langle {0,1,1} \right\rangle [/tex]

Now that you have an equation, you can represent the tangent line as :smile: :
[tex] \vec L\left( t \right) = \left\langle {1,0,0} \right\rangle + t\left\langle {0,1,1} \right\rangle \Rightarrow \vec L\left( t \right) = \left\langle {1,t,t} \right\rangle [/tex]

Or parametrically without vector notation:
[tex] L\left( t \right) = \left\{ \begin{gathered}
x = 1 \hfill \\
y = z = t \hfill \\
\end{gathered} \right\} [/tex]

:biggrin: Which basically says:
[tex] x = 1 , \,y = z[/tex]

**Hope this helps :smile:
 
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  • #4
so what is x(t)=? y(t)=? z(t)=? because i see what u did but the computer say its wrong... so i dontk now where it went wrong... i know what u did i did the same..
 
  • #5
weckod said:
so what is x(t)=? y(t)=? z(t)=? because i see what u did but the computer say its wrong... so i dontk now where it went wrong... i know what u did i did the same..
[tex] \left\{ \begin{gathered}
x\left( t \right) = 1 \hfill \\
y\left( t \right) = t \hfill \\
z\left( t \right) = t \hfill \\
\end{gathered} \right\} [/tex]

What's the problem?
 
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  • #6
well x(t) is not 1+t and same w/the others.. the computer say its wrong... that's what trippin me out
 
  • #7
wow now the y(t) and z(t) is right but the x(t) is still wrong...
 
  • #8
yay its just 1...
 
  • #9
thanks a lot dude! u helped a lot i hate cal 3 its just hard for me for some reasons... its the vectors... i can't picture them...
 
  • #10
No problem :smile: Welcome to PF
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What are parametric equations?

Parametric equations are a set of equations that define a relationship between two or more variables. They are often used to describe the path of a point or object in a coordinate system.

How do you find the tangent line at a specific point using parametric equations?

To find the tangent line at a specific point using parametric equations, you can use the derivative of the parametric equations with respect to the independent variable. This will give you the slope of the tangent line at that point, which can then be used to find the equation of the line.

What is the significance of the point (cos 0pi/6, sin 0pi/6, 0pi/6) in the context of parametric equations?

The point (cos 0pi/6, sin 0pi/6, 0pi/6) represents a specific point on a curve described by parametric equations. It is the point of interest for finding the tangent line at that particular point.

Can parametric equations be used to describe three-dimensional curves?

Yes, parametric equations can be used to describe curves in three-dimensional space. In this case, the equations will have three parameters and will describe the relationship between x, y, and z coordinates of a point on the curve.

How can parametric equations be used to model real-world phenomena?

Parametric equations can be used to model real-world phenomena by describing the relationship between two or more variables. This can be useful in fields such as physics, engineering, and economics where the behavior of a system can be represented by a set of equations.

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