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Trigonometry Solve sin³(x) - cos³(x) = sin²(x)

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,756
Hi MHB,

This seemingly very simple and straightforward trigonometric equation (Solve for x for which sin³x-cos³x=sin²x) has me stumped! I am fretting after I exhausted all kind of tricks that I could think of (which I will mention in my attempts next) trying to attempt the problem and now, I wanted so much to ask for help here at MHB!

First attempt:

I first divided through the equation by sin³x, bearing in mind sin³x≠0 and obtained:

\(\displaystyle 1-\cot^3x=cscx\)

\(\displaystyle (1-\cot^3x)^2=\csc^2x\)

\(\displaystyle (1-\cot^3x)^2=1+\cot^2x\)

If l let \(\displaystyle \cot x=k,\) the equation above becomes \(\displaystyle (1-k^3)^2=1+k^2\) and this is a futile attempt to say the least.

Second attempt:

I applied the triple angle formulas for both sine and cosine to the given equation and ended up with

\(\displaystyle \frac{3\sin x-\sin 3x}{4}-\frac{\cos3x+3\cos x}{4}=\sin^2x\)

\(\displaystyle 3\sin x-\sin 3x-\cos 3x-3\cos x=2(2\sin^2x)\)

\(\displaystyle 3(\sin x-\cos x)-(\sin3x+\cos3x)=2(1-\cos2x)
\)
This is another unwise substitution to make, knowing it can hardly bring us any closer to the answer...

Anyway, this is how I approached it.

\(\displaystyle 3( \sin x- \cos x)-(\sin3x+\cos3x)=2(1-\cos2x)\)

\(\displaystyle 3\sqrt{2}\sin(x-\frac{\pi}{4}-\sqrt{2}\sin(3x+\frac{\pi}{4})=2(1-\cos 2x)\) I stopped right at this point, because we can tell this is a very bad move...

Third attempt:

\(\displaystyle \sin^3 x-\cos^3 x=\sin^2 x\)

\(\displaystyle \sin^2 x(\sin x-1)=\cos x\cos^2 x=\cos x(1-\sin^2 x)=\cos x(1+\sin x)(1-\sin x)\)

\(\displaystyle \sin^2 x(\sin x-1)-\cos x(1+\sin x)(1-\sin x)=0\)

\(\displaystyle \sin^2 x(\sin x-1)+\cos x(1+\sin x)(\sin x-1)=0\)

\(\displaystyle (\sin^2 x+\cos x(1+\sin x))(\sin x-1)=0\)

\(\displaystyle (\sin^2 x+\cos x+\sin x\cos x))(\sin x-1)=0\)

and obviously \(\displaystyle \sin x=1\) is an answer to the problem.

And now, all kind of thoughts enter my mind, I am not even sure when we have the case ab=0, if a=0 is imminent and true, then b isn't necessary a zero. And everything seems to reach an impasse after getting sinx=1 as one of the answers to the problem.

So, I decided to factor the LHS of the given equation and see how far I can go with this approach...

\(\displaystyle (\sin x-\cos x)(\sin^2 x+\sin x\cos x+\cos^2 x)=\sin^2 x\)

\(\displaystyle (\sin x-\cos x)(1+\sin x\cos x)=\sin^2 x\)

and it's also safe to say at this juncture this won't work out well and I am at my wit's end to solve this problem.

I would appreciate it if someone could offer me some hints to solve it.

Thanks.
 

issacnewton

Member
Jan 30, 2012
61
Re: Solve sin³x-cos³x=sin²x

Hello
This is how I did it. We have,
\[ \sin^3x - \cos^3 x = \sin^2 x \]
Write this as the following.
\[ \sin^3x-\sqrt{(1-\sin^2x)}\;(1-\sin^2x) = \sin^2x \]
Here let \( t = \sin x\), and then rearranging the terms we get a polynomial in t.
\( (t^3 - t^2)^2 = (1-t^2)^3 \). Expanding this we have,
\[ 2t^6 - 2t^5 - 2t^4 + 3t^2 - 1 = 0 \]
Now using the rational root theorem and synthetic division, we can arrive at \( (t-1)^2 \) as
one factor. So obviously \(t=1\) is one root. The remainder polynomial equation then is
\[ 2t^4+2t^3 - 2t -1 = 0 \]
This is a quartic equation, for which there are known solutions. For this equation there
are two real and two imaginary solutions. The real solutions are \(t = 0.903408 \) and
\(t = -0.582522 \). Plugging back in the original equation for \( \sin x\), we can solve for \(x\).
\( x = 1.12765 \) and \( x = 3.76342 \), also the first solution of \( t=1 \) leads to \( x = \frac{\pi}{2}\).
So three solutions are \(x = 1.12765\;\;x = 3.76342\;\;x = \frac{\pi}{2} \)
 

dwsmith

Well-known member
Feb 1, 2012
1,673
Re: Solve sin³x-cos³x=sin²x

Hello
This is how I did it. We have,
\[ \sin^3x - \cos^3 x = \sin^2 x \]
Write this as the following.
\[ \sin^3x-\sqrt{(1-\sin^2x)}\;(1-\sin^2x) = \sin^2x \]
Here let \( t = \sin x\), and then rearranging the terms we get a polynomial in t.
\( (t^3 - t^2)^2 = (1-t^2)^3 \). Expanding this we have,
\[ 2t^6 - 2t^5 - 2t^4 + 3t^2 - 1 = 0 \]
Now using the rational root theorem and synthetic division, we can arrive at \( (t-1)^2 \) as
one factor. So obviously \(t=1\) is one root. The remainder polynomial equation then is
\[ 2t^4+2t^3 - 2t -1 = 0 \]
This is a quartic equation, for which there are known solutions. For this equation there
are two real and two imaginary solutions. The real solutions are \(t = 0.903408 \) and
\(t = -0.582522 \). Plugging back in the original equation for \( \sin x\), we can solve for \(x\).
\( x = 1.12765 \) and \( x = 3.76342 \), also the first solution of \( t=1 \) leads to \( x = \frac{\pi}{2}\).
So three solutions are \(x = 1.12765\;\;x = 3.76342\;\;x = \frac{\pi}{2} \)
I think you need to add \(+ 2\pi k\) where \(k\in\mathbb{Z}\) since there are infinitely many solutions for each principle argument.
 

issacnewton

Member
Jan 30, 2012
61
Re: Solve sin³x-cos³x=sin²x

I am assuming that \( x \in [0,2\pi] \)
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,725
Re: Solve sin³x-cos³x=sin²x

Hello
This is how I did it. We have,
\[ \sin^3x - \cos^3 x = \sin^2 x \]
Write this as the following.
\[ \sin^3x-\sqrt{(1-\sin^2x)}\;(1-\sin^2x) = \sin^2x \]
Here let \( t = \sin x\), and then rearranging the terms we get a polynomial in t.
\( (t^3 - t^2)^2 = (1-t^2)^3 \). Expanding this we have,
\[ 2t^6 - 2t^5 - 2t^4 + 3t^2 - 1 = 0 \]
Now using the rational root theorem and synthetic division, we can arrive at \( (t-1)^2 \) as
one factor. So obviously \(t=1\) is one root. The remainder polynomial equation then is
\[ 2t^4+2t^3 - 2t -1 = 0 \]
This is a quartic equation, for which there are known solutions. For this equation there
are two real and two imaginary solutions. The real solutions are \(t = 0.903408 \) and
\(t = -0.582522 \). Plugging back in the original equation for \( \sin x\), we can solve for \(x\).
\( x = 1.12765 \) and \( x = 3.76342 \), also the first solution of \( t=1 \) leads to \( x = \frac{\pi}{2}\).
So three solutions are \(x = 1.12765\;\;x = 3.76342\;\;x = \frac{\pi}{2} \)
I agree with the solutions $x = 3.76342$ and $x = \frac{\pi}{2}$, but not with $x = 1.12765$. The reason is that when going from $\sin x$ to $x$ you need to ensure that you choose the angle in the correct quadrant. When you check against the original equation $\sin^3x - \cos^3x = \sin^2x$, you find that both solutions $\sin x = 0.903408$ and $\sin x = -0.582522$ require $\cos x$ to be negative. So in both cases you need to have $x$ in the interval $\bigl[\frac\pi2,\frac{3\pi}2\bigr]$. Therefore the solution is not $x = 1.12765$ but $x = \pi - 1.12765 = 2.01394$.

Ideally, it would be far better to have an exact solution rather than these numerical approximations. But I am not convinced that that can be done for this problem.
 

issacnewton

Member
Jan 30, 2012
61
Re: Solve sin³x-cos³x=sin²x

Opalg, thanks for pointing mistake. I did the math in hurry.
Since you are from UK, you must have heard of S.L. Loney who did write a book
on trigonometry. I think this problem could be from that book.