# Thread: Tangent to the curve from a parametric equation

1. ## Tangent to the curve from a parametric equation

Hello,

I've got this question I've been labouring over.

The parametric equations of a curve are : x = 1 + 2sin^2Ɵ, y = 4tanƟ.
From a previous question I found dy/dx to be 1 / (sinƟcos^3Ɵ).

Now it says

"Find the equation of the tangent to the curve at the point where Ɵ = π/4, giving your answer in the form y = mx + c.
So what I did was first try get a value for x and y by plugging in π/4 in the parametric equations.

x = 1 + (sin/4)^2
x = 1.00

y = 4tan(π/4)
y = 0.05

Then I put in π/4 into the dy/dx equation.

1 / (sinπ/4)((cosπ/4)^3)
= 72.95

Using the point-slope rule:

y - y1 = m(x - x1)
y - 0.05 = 72.95(x - 1)
y = 72.95x - 72.95 + 0.05
y = 72.95x + 73

According to the mark scheme, that is WAY off, as their answer is y = 4x - 4. Where did I go wrong? Sorry if this is in the wrong category.

2. ## Re: Tangent to the curve from a parametric equation

You made several computational mistakes. Were you using a calculator perhaps? This problem should be solved without one.

When $\displaystyle \theta = \pi/4$:
$\displaystyle x(\theta) = x(\pi/4) = 1+2\sin^2(\pi/4) = 1 + 2 \left(\frac{\sqrt{2}}{2}\right)^2 = 1 + 2\left(\frac{2}{4}\right) = 1 + 1 = 2$.

$\displaystyle y(\theta) = y(\pi/4) = 4\tan(\pi/4) = 4(1) = 4$.

$\displaystyle \frac{dy}{dx}(\theta) = \frac{dy}{dx}(\pi/4) = \frac{1}{(\sin(\pi/4)\cos^3(\pi/4)} = \frac{1}{(\sqrt{2}/2)(\sqrt{2}/2)^3}$.

$\displaystyle = \frac{1}{(\sqrt{2}/2)^4} = \frac{1}{(\sqrt{16}/16)} = \frac{1}{(4/16)} = \frac{1}{(1/4)} = 4$.

So when $\displaystyle \theta = \pi/4$, it's at the point $\displaystyle (2, 4)$ and has tangent line of slope $\displaystyle 4$.

3. ## Re: Tangent to the curve from a parametric equation

Wow, you lost me.

How did you get from $\displaystyle 2\sin^2(\pi/4)$ to $\displaystyle 2 \left(\frac{\sqrt{2}}{2}\right)^2$ ?

4. ## Re: Tangent to the curve from a parametric equation

Another approach to find the slope of the tangent line:

$\displaystyle \frac{dy}{dx}=\frac{dy}{d\theta}\cdot\frac{d\theta }{dx}=\left(4\sec^2(\theta) \right)\left(\frac{1}{2\sin(2\theta)} \right)=\frac{2\sec^2(\theta)}{\sin(2\theta)}$

Hence:

$\displaystyle \frac{dy}{dx}\left|_{\theta=\frac{\pi}{4}}=\frac{4 }{1}=4$

We could also eliminate the parameter $\displaystyle \theta$, to find the equivalent Cartesian equation:

$\displaystyle x=1+2\sin^2\left(\tan^{-1}\left(\frac{y}{4} \right) \right)=1+\frac{2y^2}{y^2+16}$

$\displaystyle xy^2+16x=3y^2+16$

Implicitly differentiate with respect to $\displaystyle x$:

$\displaystyle x\cdot2y\cdot\frac{dy}{dx}+y^2+16=6y\frac{dy}{dx}$

$\displaystyle \frac{dy}{dx}=\frac{y^2+16}{2y(3-x)}$

When $\displaystyle \theta=\frac{\pi}{4}$ we have $\displaystyle (x,y)=(2,4)$ hence, at this point, we have:

$\displaystyle \frac{dy}{dx}=\frac{4^2+16}{2\cdot4(3-2)}=\frac{32}{8}=4$

5. ## Re: Tangent to the curve from a parametric equation

Originally Posted by yorkey
Wow, you lost me.

How did you get from $\displaystyle 2\sin^2(\pi/4)$ to $\displaystyle 2 \left(\frac{\sqrt{2}}{2}\right)^2$ ?
It seems you don't know that $\displaystyle \sin(\pi/4) = \frac{\sqrt{2}}{2}$.

If that's so, then you don't understand some pretty basic trigonometry. If you're taking a calculus class, then that's going to cause you serious problems. I'd suggest you take measures - self-study, a tutor, whatever - to get caught up with trigonometry.

6. ## Re: Tangent to the curve from a parametric equation

The trigonometric rules in my textbook I know and understand. They are:

y = sinx
dy/dx = cosx

y = cosx
dy/dx = -sinx

y = tanx
dy/dx = sec^2x

But it seems that they've thrown me something they haven't taught me. I'm just asking what is that rule you're talking about. Is there a more general law for it?

7. ## Re: Tangent to the curve from a parametric equation

Originally Posted by yorkey
The trigonometric rules in my textbook I know and understand. They are:

y = sinx
dy/dx = cosx

y = cosx
dy/dx = -sinx

y = tanx
dy/dx = sec^2x

But it seems that they've thrown me something they haven't taught me. I'm just asking what is that rule you're talking about. Is there a more general law for it?
it's called the unit circle ... you're expected to have been exposed to it prior to enrolling in a calculus course.

8. ## Re: Tangent to the curve from a parametric equation

Right, I've got my work cut out for myself. Thanks for the help, everyone. :-)