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How can I solve the equation $$\sin 3x=\cos 2x$$ for $x$ in the range $[0,\pi]$?

I already have a solution elaborated in the answer below.But is there any way of solving this without using the double and triple angle identities? (The problem arises as homework in a high school mathematics class which hasn't leant the identities yet.)

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This is $\cos(\frac\pi2-3x)=\cos(2x)$. Now, since $\cos A=\cos B\iff \exists k\in\Bbb Z, A-B=2k\pi\lor\exists h\in\Bbb Z, A+B=2h\pi$, this becomes $$\frac\pi2-3x=2k\pi+2x\lor \frac\pi2-3x=2h\pi-2x\\ x=-\frac25\pi k+\frac\pi{10}\lor x=-2h\pi+\frac\pi2$$

Now, the only such angles in $[0,\pi]$ are obtained for $k=0,-1,-2$ or for $h=0$. Therefore the solution is $x\in\left\{\frac\pi{10},\frac\pi2,\frac9{10}\pi\right\}$

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Yes, there is, using the a basic identity; rewrite the equation as $$\cos\Bigl(\frac\pi 2- 3x\Bigr)=\cos 2x\iff \frac\pi 2- 3x\equiv\pm 2x\pmod{2\pi}\iff\begin{cases} 5x\equiv \frac\pi2\\\phantom{5}x\equiv\frac\pi 2 \end{cases}\pmod{2\pi} $$ The first equation is equivalent to $$x\equiv\frac\pi{10}\pmod{\tfrac{2\pi}5}\quad\text{i.e.}\quad x=\frac{(4k+1)\pi}{10},\; k\in\mathbf Z.$$ There remains to sort the solutions which live in $[0,\pi]$.

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  • $\begingroup$ (+1) Nice, I wouldn't have thought of that. $\endgroup$ – YiFan Sep 29 '19 at 22:30
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From the triple angle identity for sine and double angle identity for cosine, the equation reduces to $$3\sin x-4\sin^3x=1-2\sin^2 x.$$ Naturally we substitute $s=\sin x$, then this becomes the cubic equation $$4s^3-2s^2-3s+1=0.$$ Now, observe that $s=1$ is a root, so factor that out from the LHS to get $$(s-1)(4s^2+2s-1)=0.$$ Solving the quadratic, we have $s=\frac14(-1\pm\sqrt5)$. Correspondingly, we obtain the $x$ values: $$x=\frac\pi2, \arcsin\left(\frac14(-1+\sqrt5)\right), \pi-\arcsin\left(\frac14(-1+\sqrt5)\right)=\frac\pi2,\frac\pi{10},\frac{9\pi}{10}$$

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  • $\begingroup$ No, you don't just take the $\arcsin$ of these values, because there is no $x\in [0,\pi]$ such that $\sin x=\frac{-1-\sqrt 5}4$. You take all the solutions in $[0,\pi]$ to the equations $\sin x=\alpha_i$ where $\alpha_i$ ranges over the non-negative roots of that polynomial. Namely, you'll take $\arcsin 1$, $\arcsin\frac{\sqrt 5-1}4$ and $\pi-\arcsin\frac{\sqrt 5-1}4$. $\endgroup$ – Gae. S. Sep 29 '19 at 22:24
  • $\begingroup$ @Gae.S. You're right, I have edited the answer to reflect that. Thanks for the help! $\endgroup$ – Lucas Sep 29 '19 at 22:57

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