MHB Solution to Trigonometric System

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The discussion focuses on finding real values of x, y, and z within the interval [0, π/2] that satisfy the trigonometric equations sin(x)cos(y) = sin(z) and cos(x)sin(y) = cos(z). Participants evaluate different approaches to solving the system, with one user acknowledging another's method while pointing out an oversight in identifying all possible solutions. The conversation highlights the collaborative nature of problem-solving in mathematics, as users share insights and corrections. Ultimately, the aim is to identify all valid pairs of solutions to the given equations. The thread emphasizes the importance of thoroughness in mathematical problem-solving.
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Find all reals $$x,\,y,\,z \in \left[0,\,\frac{\pi}{2}\right]$$ that satisfying the system below:

$\sin x \cos y=\sin z\\\cos x \sin y=\cos z$
 
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I can find only one solution:

Taking a square and summing up will get

$$\begin{align*}\sin^2 z + \cos^2 z = 1 &= \sin^2x\cos^2y + \cos^2x\sin^2y \\
& = \sin^2x + \sin^2y \left( \cos^2x - \sin^2x \right) \qquad \Rightarrow \end{align*} $$

$$\sin^2y = \frac{\cos^2x}{\cos^2x - \sin^2x} \ge 1.$$

Equality takes place when $$x = 0$$ and thus $$y = \frac{\pi}{2}$$ and $$z = 0$$ is the only solution.
 
We have $(\sin x \cos y)^2+(\cos x \sin y)^2 =1$\
$(\sin x \cos y)^2 <= (\sin x)^2$ and $(\sin x \cos y)^2 <= (\cos y)^2$
so on we must have
either$(\sin x \cos y) = (\sin x)$ and $(\cos x \sin y) = \cos x$ $=>\sin x = 0,\sin y = 1$ or $\cos y = 1,\cos x=0$
This gives
1) $\sin x = 0,\sin y = 1$ => $x=0, y = \frac{\pi}{2}, z= 0$
2) $\cos x = 0,\cos y = 1$ => $x=z = \frac{\pi}{2}, y= 0$

Or $(\sin x \cos y) = (\cos y )$ and $(\cos x \sin y) = \sin y$ $=>\cos x = 1,\cos y = 0$ or $\sin y = 1,\cos x=1$
THis does not gives any additional solutions
so the solution set is
3)$x=0, y = \frac{\pi}{2}, z= 0$ or $x=z = \frac{\pi}{2}, y= 0$

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Theia said:
I can find only one solution:

Taking a square and summing up will get

$$\begin{align*}\sin^2 z + \cos^2 z = 1 &= \sin^2x\cos^2y + \cos^2x\sin^2y \\
& = \sin^2x + \sin^2y \left( \cos^2x - \sin^2x \right) \qquad \Rightarrow \end{align*} $$

$$\sin^2y = \frac{\cos^2x}{\cos^2x - \sin^2x} \ge 1.$$

Equality takes place when $$x = 0$$ and thus $$y = \frac{\pi}{2}$$ and $$z = 0$$ is the only solution.
Hello Theia;
your approach is better than my approach but unfortunately you overlooked
$\sin \,y = \cos\, x = 0$
 
Hi Theia!

Albeit you missed out another pair of solution, you still deserve a pat on the back for the job well done...and thanks for participating!

My solution:

Adding up the two equations gives:

$\sin z+\cos z=\sin x \cos y+\cos x \sin y$

Now, by applying the Cauchy-Schwarz inequality to the RHS of equation above yields:

$\sin z+\cos z=\sin x \cos y+\cos x \sin y\le \sqrt{\sin^2 x+\cos^2 x} \sqrt{\sin^2 y+\cos^2 y}=1$

$\left(\sin z+\cos z\right)^2\le 1^2$

$\sin^2 z+\cos^2 z+2\sin z \cos z\le 1^2$

$\sin z \cos z\le 0$ (*)

But since we're told $$x,\,y,\,z \in \left[0,\,\frac{\pi}{2}\right]$$, the inequality (*) holds iff $\sin z=0$ or $\cos z=0$, which lead to two set of solutions where:

$$\left(x,\,y,\,z\right)=\left(0,\,\frac{\pi}{2},\,0\right),\,\left(\frac{\pi}{2},\,0,\,\frac{\pi}{2}\right)$$
 
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