If m and n respectively are the numbers of positive and negative value of θ in the interval [-π, π]

Q: Wait, why can we just cancel $\cos \frac{3\theta}{2}$ from both sides? Is that always allowed?

Yes, it is perfectly valid here because we are simply subtracting the same term from both sides of the equation. It is not like dividing by a variable expression, which could be zero and lead to lost solutions. Here, we are just simplifying the equation $A + C = B + C$ to $A = B$.

Q: Why is the general solution $\theta = \frac{k\pi}{5}$ instead of just $\frac{2k\pi}{5}$? Did I miss something?

Great catch! When you solve $\frac{15\theta}{2} = 2k\pi \pm \frac{5\theta}{2}$, you get two cases. The positive case gives $\theta = \frac{2k\pi}{5}$, and the negative case gives $\theta = \frac{k\pi}{5}$. Since $\frac{2k\pi}{5}$ is actually a subset of $\frac{k\pi}{5}$ (when $k$ is even), the set of all solutions is covered by $\theta = \frac{k\pi}{5}$.

Q: Why do we exclude $k=0$ when counting $m$ and $n$?

The question specifically asks for the number of *positive* and *negative* values of $\theta$. When $k=0$, $\theta = 0$, which is neither positive nor negative. Therefore, we exclude $k=0$ from our counts for $m$ and $n$.

Question

If $m$ and $n$ respectively are the numbers of positive and negative value of $	heta$ in the interval $[-\pi, \pi]$ that satisfy the equation $\cos 2	heta \cos \frac{	heta}{2} = \cos 3	heta \cos \frac{9	heta}{2}$, then $mn$ is equal to _____.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, fellow traveler of the JEE path. Today, we are not just solving an equation; we are uncovering a hidden symmetry in the world of trigonometry.

The problem before us is:
$$\cos 2	heta \cos \frac{	heta}{2} = \cos 3	heta \cos \frac{9	heta}{2}$$

At first glance, it looks like a chaotic mess of angles. But remember, in the realm of JEE Advanced, chaos is often just order in disguise.

### The Transformation

We start with the equation $\cos 2	heta \cos \frac{	heta}{2} = \cos 3	heta \cos \frac{9	heta}{2}$. When you see products of cosines, your mind should immediately race to the product-to-sum identities.

To use them, we need a factor of $2$. We multiply both sides by $2$:
$$2 \cos 2	heta \cos \frac{	heta}{2} = 2 \cos 3	heta \cos \frac{9	heta}{2}$$

Now, we apply the identity $2 \cos A \cos B = \cos(A+B) + \cos(A-B)$.

On the left-hand side, let $A = 2	heta$ and $B = \frac{	heta}{2}$. Adding them gives $\frac{5	heta}{2}$, and subtracting them gives $\frac{3	heta}{2}$. Thus, the left side becomes:
$$\cos \frac{5	heta}{2} + \cos \frac{3	heta}{2}$$

On the right-hand side, let $A = \frac{9	heta}{2}$ and $B = 3	heta$. Adding them gives $\frac{15	heta}{2}$, and subtracting them gives $\frac{3	heta}{2}$. Thus, the right side becomes:
$$\cos \frac{15	heta}{2} + \cos \frac{3	heta}{2}$$

### The Elegant Cancellation

Look at what we have now:
$$\cos \frac{5	heta}{2} + \cos \frac{3	heta}{2} = \cos \frac{15	heta}{2} + \cos \frac{3	heta}{2}$$

The term $\cos \frac{3	heta}{2}$ is waiting on both sides like a silent observer. We subtract it away, and suddenly, the equation collapses into something beautiful:
$$\cos \frac{15	heta}{2} = \cos \frac{5	heta}{2}$$

This is the moment where the complexity vanishes, leaving us with a clean, solvable structure.

### The General Solution

We know that the general solution for $\cos X = \cos \alpha$ is $X = 2k\pi \pm \alpha$, where $k \in \mathbb{Z}$. Applying this to our equation, we get:
$$\frac{15	heta}{2} = 2k\pi \pm \frac{5	heta}{2}$$

This splits into two cases.

**Case 1 (the positive sign):**
$$\frac{15	heta}{2} = 2k\pi + \frac{5	heta}{2} \Rightarrow \frac{10	heta}{2} = 2k\pi \Rightarrow 5	heta = 2k\pi \Rightarrow 	heta = \frac{2k\pi}{5}$$

**Case 2 (the negative sign):**
$$\frac{15	heta}{2} = 2k\pi - \frac{5	heta}{2} \Rightarrow \frac{20	heta}{2} = 2k\pi \Rightarrow 10	heta = 2k\pi \Rightarrow 	heta = \frac{k\pi}{5}$$

Since $\frac{2k\pi}{5}$ is just a subset of $\frac{k\pi}{5}$, our general solution is simply:
$$	heta = \frac{k\pi}{5}$$

### Final Calculation

We are restricted to the interval $[-\pi, \pi]$. Setting $-\pi \le \frac{k\pi}{5} \le \pi$, we find that $k$ must be an integer in the range $[-5, 5]$.

The possible values for $k$ are $\{-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}$.

The positive values of $	heta$ occur for $k \in \{1, 2, 3, 4, 5\}$, so $m = 5$. The negative values of $	heta$ occur for $k \in \{-1, -2, -3, -4, -5\}$, so $n = 5$.

The product $mn$ is:
$$5 	imes 5 = \mathbf{25}$$

We have navigated the storm and arrived at the shore. The beauty of this problem lies not in the final number, but in the elegance of the path we took to find it.

Animated Solution for Mathematics - Trigonometry: If $m$ and $n$ respectively are the numbers of positive and negative value of $θ$ in the interval $[- π, π]$ that satisfy the equation $cos 2 θ cos \frac{θ}{2} = cos 3 θ cos \frac{9 θ}{2}$ , then $mn$ is equal to _____.

Enter Numerical Value:

Visualized Solution

Initial Equation Setup

Applying Product-to-Sum Identity

Simplifying the Equation

General Solution for Cosine

Case 1: Positive Sign

Case 2: Negative Sign

Visualizing the Interval

Finding Positive Values ( $m$ )

Finding Negative Values ( $n$ )

Final Calculation of $mn$

The Sigma Insight: General Solution of Trigonometric Equations

Analyzing the Setup

The Transformation

The Elegant Cancellation

The General Solution

Final Calculation

Similar Questions

The number of values of $θ$ in the interval $(- π /2, π /2)$ such that $θ \neq = \frac{nπ}{5}$ for $n = 0, \pm 1, \pm 2$ and $tan θ = cot 5 θ$ as well as $sin 2 θ = cos 4 θ$ is ____.

The number of solutions of the equation $cos 2 θ cos \frac{θ}{2} + cos \frac{5 θ}{2} = 2 cos^{3} \frac{5 θ}{2}$ in $[- \frac{π}{2}, \frac{π}{2}]$ is :

The sum of all values of $θ \in [0, 2 π]$ satisfying $2 sin^{2} θ = cos 2 θ$ and $2 cos^{2} θ = 3 sin θ$ is

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

If $θ_{1}$ and $θ_{2}$ be respectively the smallest and the largest values of $θ$ in $(0, 2 π) - {π}$ which satisfy the equation, $2 cot^{2} θ - \frac{5}{s i n θ} + 4 = 0$ , then $\int_{θ_{1}}^{θ_{2}} cos^{2} 3 θ d θ$ 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:

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:

The number of solutions of the pair of equations $2 sin^{2} θ - cos 2 θ = 0$ , $2 cos^{2} θ - 3 sin θ = 0$ in the interval $[0, 2 π]$ is

The number of solutions of the equation $cos (x + \frac{π}{3}) cos (\frac{π}{3} - x) = \frac{1}{4} cos^{2} 2 x$ , $x \in [- 3 π, 3 π]$ is :

The number of solutions of the equation $sin x = cos^{2} x$ in the interval $(0, 10)$ is ______.

Enter Numerical Value:

Visualized Solution

Initial Equation Setup

Applying Product-to-Sum Identity

Simplifying the Equation

General Solution for Cosine

Case 1: Positive Sign

Case 2: Negative Sign

Visualizing the Interval

Finding Positive Values ( .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } m)

Finding Negative Values ( .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n)

Final Calculation of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } mn

The Sigma Insight: General Solution of Trigonometric Equations

Analyzing the Setup

The Transformation

The Elegant Cancellation

The General Solution

Final Calculation

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of values of θ in the interval (−π/2,π/2) such that θ=5nπ​ for n=0,±1,±2 and tanθ=cot5θ as well as sin2θ=cos4θ is ____.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of solutions of the equation cos2θcos2θ​+cos25θ​=2cos325θ​ in [−2π​,2π​] is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of all values of θ∈[0,2π] satisfying 2sin2θ=cos2θ and 2cos2θ=3sinθ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of solutions of the equation 2θ−cos2θ+2​=0 in R is equal to ______.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If θ∈[−2π,2π], then the number of solutions of 22​cos2θ+(2−6​)cosθ−3​=0, is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If θ∈[−2π,2π], then the number of solutions of 22​cos2θ+(2−6​)cosθ−3​=0, is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of solutions of the pair of equations 2sin2θ−cos2θ=0, 2cos2θ−3sinθ=0 in the interval [0,2π] is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of solutions of the equation cos(x+3π​)cos(3π​−x)=41​cos22x, x∈[−3π,3π] is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of solutions of the equation sinx=cos2x in the interval (0,10) is ______.

Finding Positive Values ( $m$ )

Finding Negative Values ( $n$ )

Final Calculation of $mn$

The number of values of $θ$ in the interval $(- π /2, π /2)$ such that $θ \neq = \frac{nπ}{5}$ for $n = 0, \pm 1, \pm 2$ and $tan θ = cot 5 θ$ as well as $sin 2 θ = cos 4 θ$ is ____.

The number of solutions of the equation $cos 2 θ cos \frac{θ}{2} + cos \frac{5 θ}{2} = 2 cos^{3} \frac{5 θ}{2}$ in $[- \frac{π}{2}, \frac{π}{2}]$ is :

The sum of all values of $θ \in [0, 2 π]$ satisfying $2 sin^{2} θ = cos 2 θ$ and $2 cos^{2} θ = 3 sin θ$ is

The number of solutions of the equation $2 θ - cos^{2} θ + 2 = 0$ in $R$ 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:

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:

The number of solutions of the pair of equations $2 sin^{2} θ - cos 2 θ = 0$ , $2 cos^{2} θ - 3 sin θ = 0$ in the interval $[0, 2 π]$ is

The number of solutions of the equation $cos (x + \frac{π}{3}) cos (\frac{π}{3} - x) = \frac{1}{4} cos^{2} 2 x$ , $x \in [- 3 π, 3 π]$ is :

The number of solutions of the equation $sin x = cos^{2} x$ in the interval $(0, 10)$ is ______.