The number of all possible values of θ where 0

Q: Wait, why can we just equate the first and third equations? Is that always allowed?

Yes, it is! Since both equations are equal to the same expression, $(xyz) \sin 3\theta$, they are essentially two different ways of describing the same value. By setting them equal, we are simply using the transitive property of equality ($a=b$ and $a=c \implies b=c$) to eliminate the common term and focus on the relationship between the other variables.

Q: How do we know for sure that $x$ must be zero?

Great question. When we simplified the second equation, we found $x \sin 3\theta = \frac{2(z \cos 3\theta + y \sin 3\theta)}{yz}$. Because we already established from the first and third equations that $z \cos 3\theta + y \sin 3\theta = 0$, the entire numerator becomes zero. Since $\sin 3\theta$ cannot be zero (as that would force $z=0$, violating the problem's condition), $x$ must be zero to satisfy the equation.

Q: Why do we have to check if $\cos 3\theta = 0$?

In mathematics, whenever you divide by a variable or a function, you must ensure you are not dividing by zero. If $\cos 3\theta$ were zero, our derived relation $-z \cos 3\theta = y \sin 3\theta$ would imply $y \sin 3\theta = 0$. Since $\cos 3\theta = 0$ implies $\sin 3\theta = \pm 1$, this would force $y=0$, which contradicts the given condition $y_0 z_0 \neq 0$. Thus, $\cos 3\theta$ cannot be zero.

Question

The number of all possible values of $	heta$ where $0 < 	heta < \pi$, for which the system of equations $(y+z) \cos 3	heta = (xyz) \sin 3	heta$, $x \sin 3	heta = \frac{2 \cos 3	heta}{y} + \frac{2 \sin 3	heta}{z}$, $(xyz) \sin 3	heta = (y+2z) \cos 3	heta + y \sin 3	heta$ have a solution $(x_0, y_0, z_0)$ with $y_0 z_0 
eq 0$, is ____.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine standing before a complex, multi-layered system of equations. It feels like a labyrinth, but in the world of JEE Advanced, complexity is often just a mask for elegance.

We have three equations involving trigonometric functions of $3	heta$ and variables $x, y,$ and $z$. Our journey begins by observing the first and third equations:

$$(y+z) \cos 3	heta = (xyz) \sin 3	heta$$

$$(xyz) \sin 3	heta = (y+2z) \cos 3	heta + y \sin 3	heta$$

The golden key is that both equations share the exact same term, $(xyz) \sin 3	heta$. By equating them, we perform an act of mathematical liberation, removing the $x$ variable entirely and creating a bridge between $y, z,$ and $	heta$.

### The Algebraic Collapse

By setting $(y+z) \cos 3	heta = (y+2z) \cos 3	heta + y \sin 3	heta$, we begin the process of simplification. Expanding both sides gives us:

$$y \cos 3	heta + z \cos 3	heta = y \cos 3	heta + 2z \cos 3	heta + y \sin 3	heta$$

Notice the $y \cos 3	heta$ on both sides? They vanish, leaving us with $z \cos 3	heta = 2z \cos 3	heta + y \sin 3	heta$.

Rearranging this, we arrive at the beautiful, compact relation:

$$-z \cos 3	heta = y \sin 3	heta$$

This is the heart of the problem. Dividing both sides by $y \cos 3	heta$, we find that:

$$	an 3	heta = -\frac{z}{y}$$

### The Second Equation Reveal

Now, we turn our attention to the second equation:

$$x \sin 3	heta = \frac{2 \cos 3	heta}{y} + \frac{2 \sin 3	heta}{z}$$

Finding a common denominator, we get:

$$x \sin 3	heta = \frac{2z \cos 3	heta + 2y \sin 3	heta}{yz}$$

Look closely at that numerator. It is exactly twice our previous relation, $-z \cos 3	heta = y \sin 3	heta$, which implies $z \cos 3	heta + y \sin 3	heta = 0$.

Because the numerator is zero, we are left with the conclusion that $x \sin 3	heta = 0$. Since $\sin 3	heta$ cannot be zero (which would force $z=0$ and violate the system), we conclude that **$x=0$**.

### The Final Synthesis

With $x=0$, the first equation simplifies to $(y+z) \cos 3	heta = 0$. Since $\cos 3	heta 
eq 0$, we must have $y+z=0$, or **$y = -z$**.

Now, we return to our tangent relation: $	an 3	heta = -\frac{z}{y}$. Substituting $y = -z$, we get:

$$	an 3	heta = -\frac{z}{-z} = 1$$

Given $0 < 	heta < \pi$, we have $0 < 3	heta < 3\pi$. The solutions for $3	heta$ are $\frac{\pi}{4}, \frac{5\pi}{4},$ and $\frac{9\pi}{4}$.

Dividing by 3, we find the final values for $	heta$:

$$	heta = \frac{\pi}{12}, \frac{5\pi}{12}, \frac{3\pi}{4}$$

We have navigated the labyrinth and found exactly **3 solutions**.

Enter Numerical Value:

Visualized Solution

Analyze the System of Equations

Equate Equations 1 and 3

Simplify to Find Relation

Establish the Tangent Relation

Analyze the Second Equation

Deduce that $x = 0$

Find Relation Between $y$ and $z$

Solve for $tan 3 θ$

Graphical Setup for Solutions

Draw the Curves

Find the Number of Solutions

The Sigma Insight: General Solution of Trigonometric Equations

Analyzing the Setup

The Algebraic Collapse

The Second Equation Reveal

The Final Synthesis

Similar Questions

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Enter Numerical Value:

Visualized Solution

Analyze the System of Equations

Equate Equations 1 and 3

Simplify to Find Relation

Establish the Tangent Relation

Analyze the Second Equation

Deduce that .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=0

Find Relation Between .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } y and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } z

Solve for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } tan3θ

Graphical Setup for Solutions

Draw the Curves

Find the Number of Solutions

The Sigma Insight: General Solution of Trigonometric Equations

Analyzing the Setup

The Algebraic Collapse

The Second Equation Reveal

The Final Synthesis

Similar Questions

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Solve for $tan 3 θ$

The number of solutions of the equation $2 θ - cos^{2} θ + 2 = 0$ in $R$ is equal to ______.

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If $θ \in [- 2 π, 2 π]$ , then the number of solutions of $22 cos^{2} θ + (2 - 6) cos θ - 3 = 0$ , is equal to:

If $θ \in [- 2 π, 2 π]$ , then the number of solutions of $22 cos^{2} θ + (2 - 6) cos θ - 3 = 0$ , is equal to:

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