# Find the angle made by two tangents

1. Mar 1, 2015

### Calpalned

1. The problem statement, all variables and given/known data
Find the angle made by the two tangents to the curve $x=\sin2t$ and $y=\sin(2t)\cos(2t)$ at the point $(0,0)$

2. Relevant equations
$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$
(Derivative of a parametric equation)

3. The attempt at a solution
$x = \sin(2t) = 0$ when $t = 0, \frac{\pi}{2}, \pi,...$
$y=\sin(2t)\cos(2t) = 0$ at the same values of $t$
Taking the derivative of the parametric equations by using the formula in part two, I get $\frac{2\cos^2(2t)-2t\sin^2t}{2\cos(2t)}$ I get to plug in any value of t, so I choose $t=\pi/2$
With that t value, I get $\frac{2(-1^2)-2(0)}{-2}$ which is equal to $-1$
Now I'll try $t=0$, and I get $1$. Using $t=\pi$ I also get one. Therefore, the angle must be between -1 and 1 and be equal to $t = 0, \frac{\pi}{2}, \pi,...$
My answer is 0.
The correct answer is $\frac{\pi}{2}$
Could someone please enlighten me as to my mistake? That would be highly appreciated. Thanks.

Last edited by a moderator: Mar 1, 2015
2. Mar 1, 2015

### Brian T

Think about what you calculated...
The slope you got at one time was 1, the slope you got at another time was -1 (both at the same point).
What is the angle between a line of slope 1 and a line of slope -1?

You don't want to guess that because you got -1 and 1 as answers, the best thing to do is average them to get 0. -1 and 1 have nothing directly to do with angles, those are your slopes (dy/dx)

3. Mar 1, 2015

### Calpalned

I see! The angle between them is 90 degrees. Thank you so much.

4. Mar 1, 2015

### Calpalned

Now I get it! The angle is 90 degrees. Thank you so much!

5. Mar 1, 2015

### Brian T

Glad to help. and just so you can visualize the parametrization:
http://www4b.wolframalpha.com/Calculate/MSP/MSP11162061a3322c6679i6000012f58ad84fgbh939?MSPStoreType=image/gif&s=64&w=286.&h=97.&cdf=Animation [Broken]
We have two lines that cross the origin, with the slopes that you figured out.

Last edited by a moderator: May 7, 2017
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