Finding t in the Two Tangent Lines at (0,2)

In summary, the conversation discusses finding two tangent lines at the point (0,2) for two given equations. It is mentioned that the value of t for this point is equal to pi/2 and -pi/2 and this is solved by setting x=0 and y=2. The equation is then simplified to t=+-pi/2. It is also noted that there are no points at which the curve goes through (0,2) with the addition of 2n pi.
  • #1
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Homework Statement [/b]
My book talks about find the two tangent lines at the point (0,2) for http://mathbin.net/equations/7402_0.png and http://mathbin.net/equations/7402_1.png .[/URL] It says that t then is equal to pi/2 and -pi/2. I do not know how to they solved for this t. Any help?
 
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  • #2
You solve the simultaneous equation
[tex]x = 0, y = 2[/tex]

It's easiest starting with the latter:
[tex]2 = y = 2 - \pi \cos t \implies 0 = - \pi \cos t \implies t = \pm \frac{\pi}{2}[/tex]
and then all you have to do is plug them both into the equation for x and check that it gives zero (i.e. you have two values of t for which (0, 2) is on the curve).
 
  • #3
Why didn't we include [tex]\frac{ \pi }{2}+2n \pi[/tex]?

Thank you!
 
  • #4
Because there are no such points at which the curve goes through (0, 2).
You can plug it in:
x(pi/2 + 2 pi) = ... ?
 

1. What is the significance of finding t in the two tangent lines at (0,2)?

Finding t in the two tangent lines at (0,2) allows us to determine the slope of the tangent lines at that specific point on the curve. This information is useful in understanding the behavior of the curve at (0,2) and can also help in solving related problems in calculus and geometry.

2. How do you find t in the two tangent lines at (0,2)?

To find t in the two tangent lines at (0,2), we can use the point-slope form of a line to write out the equations for the two tangent lines. Then, we can set the equations equal to each other and solve for t. Alternatively, we can use the derivative of the curve at (0,2) to find the slope of the tangent lines and then use the point-slope form to write out the equations and solve for t.

3. What is the relationship between t and the slope of the tangent lines at (0,2)?

The value of t represents the slope of the tangent lines at (0,2). This means that the higher the value of t, the steeper the slope of the tangent lines will be, and vice versa. Additionally, a positive value of t indicates an increasing slope, while a negative value of t indicates a decreasing slope.

4. Can t have multiple values for the two tangent lines at (0,2)?

Yes, t can have multiple values for the two tangent lines at (0,2). This is because there can be multiple tangent lines that pass through the point (0,2) on a curve. Each tangent line will have its own unique value of t, representing its slope at that point.

5. How can finding t in the two tangent lines at (0,2) be applied in real-world situations?

In real-world situations, finding t in the two tangent lines at (0,2) can be applied in various fields such as engineering, physics, and economics. For example, in engineering, finding t can help determine the slope of a curve representing a bridge or road at a specific point, which is crucial for ensuring structural stability. In economics, t can represent the marginal rate of change of a function, which is useful in analyzing the behavior of markets and economies.

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