1. ## Parametric equations

A curve is given parametrically by
$\displaystyle x=a(5\cos\theta+\cos 5\theta)$
$\displaystyle y=a(5\sin\theta-\sin 5\theta)$
Find the equation of the normal to the curve at the point with parameter $\displaystyle \theta$.
Find those points on the curve at which the normal is also normal to the curve at another point.

my attempt:
$\displaystyle \frac{dx}{d\theta}=-5a(\sin\theta+\sin 5\theta)$
$\displaystyle \frac{dy}{d\theta}=5a(\cos\theta-\cos 5\theta)$
$\displaystyle \frac{dy}{dx}=-\frac{\cos\theta-\cos 5\theta}{\sin\theta+\sin 5\theta}$
normal
$\displaystyle \frac{\sin\theta+\sin 5\theta}{cos\theta-\cos 5\theta}=-\frac{2\sin 3\theta\cos 2\theta}{2\sin 3\theta\sin 2\theta}=-\frac{\cos 2\theta}{\sin 2\theta}$
equation
$\displaystyle y-a(5\sin\theta-\sin 5\theta)=-\frac{\cos 2\theta}{\sin 2\theta}(x-a(5\cos\theta+\cos 5\theta))$
$\displaystyle y\sin 2\theta-a\sin 2\theta(5\sin\theta-\sin 5\theta)=-x\cos 2\theta+a\cos 2\theta(5\cos\theta+\cos 5\theta)$
$\displaystyle y\sin 2\theta+x\cos 2\theta=a[5\cos\theta\cos 2\theta+\cos 2\theta\cos 5\theta+5\sin 2\theta\sin\theta-\sin 2\theta\sin 5\theta]$
$\displaystyle y\sin 2\theta+x\cos 2\theta=$$\displaystyle a[\frac{5}{2}(\cos 3\theta+\cos\theta)+\frac{1}{2}(\cos 7\theta+\cos 3\theta)-\frac{5}{2}(\cos 3\theta-\cos\theta)+\frac{1}{2}(\cos 7\theta-\cos 3\theta)] \displaystyle y\sin 2\theta+x\cos 2\theta=a(5\cos\theta+\cos 7\theta) The answer is: \displaystyle y\sin 2\theta+x\cos 2\theta=a(5\cos\theta-\cos 3\theta) So I got the RHS correct, but where on the LHS is wrong? And for finding the points where the normal there is normal at another point, I don't know where to start. Thanks for any help 2. Hello arze Originally Posted by arze A curve is given parametrically by \displaystyle x=a(5\cos\theta+\cos 5\theta) \displaystyle y=a(5\sin\theta-\sin 5\theta) Find the equation of the normal to the curve at the point with parameter \displaystyle \theta. Find those points on the curve at which the normal is also normal to the curve at another point. my attempt: Gradient \displaystyle \frac{dx}{d\theta}=-5a(\sin\theta+\sin 5\theta) \displaystyle \frac{dy}{d\theta}=5a(\cos\theta-\cos 5\theta) \displaystyle \frac{dy}{dx}=-\frac{\cos\theta-\cos 5\theta}{\sin\theta+\sin 5\theta} normal \displaystyle \frac{\sin\theta+\sin 5\theta}{cos\theta-\cos 5\theta}=-\frac{2\sin 3\theta\cos 2\theta}{2\sin 3\theta\sin 2\theta}=-\frac{\cos 2\theta}{\sin 2\theta} equation \displaystyle y-a(5\sin\theta-\sin 5\theta)=-\frac{\cos 2\theta}{\sin 2\theta}(x-a(5\cos\theta+\cos 5\theta)) \displaystyle y\sin 2\theta-a\sin 2\theta(5\sin\theta-\sin 5\theta)=-x\cos 2\theta+a\cos 2\theta(5\cos\theta+\cos 5\theta) \displaystyle y\sin 2\theta+x\cos 2\theta=a[5\cos\theta\cos 2\theta+\cos 2\theta\cos 5\theta+5\sin 2\theta\sin\theta-\sin 2\theta\sin 5\theta] \displaystyle y\sin 2\theta+x\cos 2\theta=$$\displaystyle a[\frac{5}{2}(\cos 3\theta+\cos\theta)+\frac{1}{2}(\cos 7\theta+\cos 3\theta)-\frac{5}{2}(\cos 3\theta-\cos\theta)+\frac{1}{2}(\cos 7\theta-\cos 3\theta)]$
$\displaystyle y\sin 2\theta+x\cos 2\theta=a(5\cos\theta+\cos 7\theta)$
The answer is: $\displaystyle y\sin 2\theta+x\cos 2\theta=a(5\cos\theta-\cos 3\theta)$
So I got the RHS correct, but where on the LHS is wrong?
And for finding the points where the normal there is normal at another point, I don't know where to start.
Thanks for any help
Thanks for showing us all your detailed working. I think you are right, and the stated answer is wrong. Here's why:

The normal must (obviously) pass through the point with parameter $\displaystyle \theta$. So when $\displaystyle x = a(5\cos\theta +\cos5\theta)$ and $\displaystyle y= a(5\sin\theta-\sin5\theta)$ the value of $\displaystyle y\sin 2\theta+x\cos 2\theta$ is

$\displaystyle a(5\sin\theta-\sin5\theta)\sin2\theta + a(5\cos\theta +\cos5\theta)\cos2\theta$

$\displaystyle =5a(\sin\theta\sin2\theta+\cos\theta\cos2\theta) +a(\cos5\theta\cos2\theta-\sin5\theta\sin2\theta)$

$\displaystyle =5a\cos\theta+a\cos7\theta$

So your equation contains the original point, has the required gradient and is therefore the correct equation of the normal.

As far as the second part is concerned, you may find the following idea useful.

If this line is also normal at a point with parameter $\displaystyle \phi$, then the gradients at $\displaystyle \theta$ and $\displaystyle \phi$ are equal. So, simplifying your expression for $\displaystyle \frac{dy}{dx}$, $\displaystyle \tan2\theta = \tan2\phi$

$\displaystyle \Rightarrow 2\phi = n\pi +2\theta, n \in \mathbb{Z}$

$\displaystyle \Rightarrow \phi = \frac{n\pi}{2} +\theta, n \in \mathbb{Z}$

3. Should it be $\displaystyle -\cot 2\theta=-\cot 2\phi$? Then how to go about finding the value of $\displaystyle \theta$? I can see that values of $\displaystyle \theta$ in the first quadrant will have values for $\displaystyle \phi$ in the third quadrant, but other than that I don't know how to continue.

4. Hello arze
Originally Posted by arze
Should it be $\displaystyle -\cot 2\theta=-\cot 2\phi$? Then how to go about finding the value of $\displaystyle \theta$? I can see that values of $\displaystyle \theta$ in the first quadrant will have values for $\displaystyle \phi$ in the third quadrant, but other than that I don't know how to continue.

Yes, $\displaystyle -\cot 2\theta=-\cot 2\phi$, but I was using the rather simpler gradient of the tangent, which gives the same thing in a different format: $\displaystyle \tan 2\theta=\tan 2\phi$.

To solve the problem, I should try the following:

• find the gradient of the chord joining the points with parameters $\displaystyle \theta$ and $\displaystyle \phi$, simplifying the result using the trig identities for the sums and differences of sines and cosines;

• use the relation $\displaystyle \phi = \theta + \frac{n\pi}{2}$ for different values of $\displaystyle n$ (I think it might just be for $\displaystyle n$ even and $\displaystyle n$ odd);

• equate the result to the gradient of the normal, $\displaystyle -\cot\theta$;

• and then solve the resulting equation for $\displaystyle \theta$.

I don't have time to try this right now, but that's the only method I can think of.

5. I've been trying the method you suggested, tried it many times over but still can't solve for $\displaystyle \theta$, although I think it will work, just haven't hit on the right thing.

6. Hello arze
Originally Posted by arze
I've been trying the method you suggested, tried it many times over but still can't solve for $\displaystyle \theta$, although I think it will work, just haven't hit on the right thing.
The attached image shows what the graph looks like for $\displaystyle a = 1$ (I put the values into an Excel spreadsheet). The dots represent the values of $\displaystyle \theta$ every $\displaystyle 10^o$, or $\displaystyle \frac{\pi}{18}$ radians.

Among other things, this confirms that the answer given for the normal is incorrect. If you look, for example, where $\displaystyle \theta = 0$, the point on the graph is $\displaystyle (6,0)$, the gradient is zero, and the normal is therefore $\displaystyle x= 6$. This is consistent with our answer of $\displaystyle y\sin 2\theta+x\cos 2\theta=a(5\cos\theta+\cos 7\theta)$, but not with the given answer of $\displaystyle y\sin 2\theta+x\cos 2\theta=a(5\cos\theta-\cos 3\theta)$, which gives $\displaystyle x = 4$ when $\displaystyle \theta = 0$.

I thought I had a solution to the second part of the question, based on the method I outlined, but I have found an error which I cannot currently resolve. I think that the solutions are $\displaystyle \theta = \frac{\pi}{6},\frac{\pi}{2},\frac{5\pi}{6}, \frac{7\pi}{6},\frac{3\pi}{2},\frac{11\pi}{6}$, but cannot prove this as yet!

P.S. I have just realised that we have made a mistake in the sign of the gradient of the normal:

$\displaystyle \cos\theta - \cos5\theta=2\sin3\theta\sin2\theta$ not $\displaystyle -2\sin3\theta\sin2\theta$

So $\displaystyle \frac{\sin\theta+\sin 5\theta}{\cos\theta-\cos 5\theta}=\frac{2\sin 3\theta\cos 2\theta}{2\sin 3\theta\sin 2\theta}=\frac{\cos 2\theta}{\sin 2\theta}$

Sorry, I don't have time now to follow this up - but perhaps you can correct the working that comes after this.

$\displaystyle \cos\theta - \cos5\theta=2\sin3\theta\sin2\theta$ not $\displaystyle -2\sin3\theta\sin2\theta$
So $\displaystyle \frac{\sin\theta+\sin 5\theta}{\cos\theta-\cos 5\theta}=\frac{2\sin 3\theta\cos 2\theta}{2\sin 3\theta\sin 2\theta}=\frac{\cos 2\theta}{\sin 2\theta}$
Its actually $\displaystyle 5\cos\theta+\cos 5\theta$ and $\displaystyle 5\sin\theta-\sin 5\theta$