Quadratic Equation Edexcel F2 June 2017

[fasih178]

Q8 of FP2 IAL June 2017 paper. I was able to do parts (a) and (b), but am lost with (c). I can't understand how (c) is linked with (b).
The solution uses tan5x=0 to find the quadratic. Why?

 

[MarkFL]

I would begin as you did with:

[MATH](\cos(\theta)+i\sin(\theta))^5=\cos(5\theta)+i\sin(5\theta)[/MATH]
[MATH](\cos(\theta)+i\sin(\theta))^5=\cos^5(\theta)+5\cos^4(\theta)i\sin(\theta)+10\cos^3i^2\sin^2(\theta)+10\cos^2(\theta)i^3\sin^3(\theta)+5\cos(\theta)i^4\sin^4(\theta)+i^5\sin^5(\theta)[/MATH]
Hence:

[MATH]\cos(5\theta)+i\sin(5\theta)=(\cos^5(\theta)-10\cos^3\sin^2(\theta)+5\cos(\theta)\sin^4(\theta))+i(5\cos^4(\theta)\sin(\theta)-10\cos^2(\theta)\sin^3(\theta)+\sin^5(\theta))[/MATH]
Equating real and imaginary parts, we find:

[MATH]\cos(5\theta)=\cos^5(\theta)-10\cos^3\sin^2(\theta)+5\cos(\theta)\sin^4(\theta)[/MATH]
[MATH]\sin(5\theta)=5\cos^4(\theta)\sin(\theta)-10\cos^2(\theta)\sin^3(\theta)+\sin^5(\theta)[/MATH]
And so we may write:

[MATH]\tan(5\theta)=\frac{\sin(5\theta)}{\cos(5\theta)}\cdot\frac{\sec^5(\theta)}{\sec^5(\theta)}=\frac{t^5-10t^3+5t}{5t^4-10t^2+1}[/MATH]
Where [MATH]t=\tan(\theta)[/MATH] and \(\cos(\theta)\ne0\)

Now, we know:

[MATH]\tan(\pi)=\tan(2\pi)=0[/MATH]
And so we must have (using the numerator of the above expression for \(\tan(5\theta)\)):

1.) [MATH]\tan^4\left(\frac{\pi}{5}\right)-10\tan^2\left(\frac{\pi}{5}\right)+5=0[/MATH]
2.) [MATH]\tan^4\left(\frac{2\pi}{5}\right)-10\tan^2\left(\frac{2\pi}{5}\right)+5=0[/MATH]
Can you now see the quadratic having the required roots?

 

[fasih178]

Thank you! I understood one crucial thing: we need to substitute in pie into LHS tan(5theta) so that we get theta=pie/5, which is one of our roots. And, since tan(pie) = 0, we can set our RHS expression = 0 too. By the same intuition, tan(2pie) is zero. Thus, overall, I understood the idea behind setting tan(5theta) = 0.

But what I don't get is why did you even think about tan(pie) = 0 in the first place -- then going on to using this to solve the problem. That is, you could have instead chosen, say, tan(pie/4) = 1, but you didn't. Why not? What gave you the idea to use tan(pie) = 0?

 

[MarkFL]

Thank you! I understood one crucial thing: we need to substitute in pie into LHS tan(5theta) so that we get theta=pie/5, which is one of our roots. And, since tan(pie) = 0, we can set our RHS expression = 0 too. By the same intuition, tan(2pie) is zero. Thus, overall, I understood the idea behind setting tan(5theta) = 0.

But what I don't get is why did you even think about tan(pie) = 0 in the first place -- then going on to using this to solve the problem. That is, you could have instead chosen, say, tan(pie/4) = 1, but you didn't. Why not? What gave you the idea to use tan(pie) = 0?

We previously derived the formula:

[MATH]\tan(5\theta)=\frac{t^5-10t^3+5t}{5t^4-10t^2+1}[/MATH]
This gives us a way to relate \(\tan(5\theta)\) to \(\tan(\theta)\). Now consider that:

[MATH]\pi=5\cdot\frac{\pi}{5}[/MATH]
[MATH]2\pi=5\cdot\frac{2\pi}{5}[/MATH]
And so if \(\theta=n\dfrac{\pi}{5}\) then \(5\theta=n\pi\).

The reason I chose values for which \(\tan(5\theta)=0\) is because we are searching for roots of a quadratic. Using the formula relating \(\tan(5\theta)\) to \(\tan(\theta)\), we see that in order for \(\tan(5\theta)=0\) we must have:

[MATH]\tan^5(\theta)-10\tan^3(\theta)+5\tan(\theta)=0[/MATH]
[MATH]\tan(\theta)\left(\tan^4(\theta)-10\tan^2(\theta)+5\right)=0[/MATH]
Now, when we have:

[MATH]0<\theta<\pi[/MATH]
We know \(\tan(\theta)\ne0\), and since both angles under consideration fall within that range, we may divide by \(\tan(\theta)\) to obtain the quadratic in \(\tan^2(\theta)\):

[MATH]\tan^4(\theta)-10\tan^2(\theta)+5=0[/MATH]
Since \(\tan(k\pi)=0\) where \(k\in\mathbb{Z}\), we know the above quadratic is true for all:

[MATH]\theta=k\frac{\pi}{5}[/MATH]
Because of the fact that on the unit circle, an angle of \(\theta\) is equivalent to \(\theta+2k\pi\), we know then that for:

[MATH]\theta\in\left\{k\frac{\pi}{5},(k+1)\frac{\pi}{5}\right\}[/MATH]
[MATH]\tan^2(\theta)[/MATH] must be distinct roots of the quadratic:

[MATH]x^2-10x+5=0[/MATH]
Now given that the product of the two roots of a quadratic is the quotient of the constant term divided by the coefficient of the squared term, we may conclude that:

[MATH]\tan^2\left(k\frac{\pi}{5}\right)\tan^2\left((k+1)\frac{\pi}{5}\right)=5[/MATH]
Since for \(k=1\) we have \(\theta\) in the first quadrant, we can therefore state:

[MATH]\tan\left(\frac{\pi}{5}\right)\tan\left(\frac{2\pi}{5}\right)=\sqrt{5}[/MATH]
Does this make sense?

 

[fasih178]

Oh yes, it does! It makes perfect sense now. However, are there any resources or videos I can watch to hone in on such questions, particularly those that pertain to making these connections using trigonometric functions? I know this was rather intuitive for you, but I have difficulty grasping such connections.

 

[MarkFL]

It really just takes practice. The more problems you work, the easier it becomes to begin making these connections.

 

[fasih178]

Indeed. The old adage ''Practice makes perfect'' applies best to Math.

Thank you.