If θ ∈ [-7π/6, 4π/3], then the number of solutions of √3 cosec² θ - 2(√3 - 1) cosec θ - 4 = 0, is eq

Q: Why do we need to convert to $\sin \theta$?

While $\operatorname{cosec} \theta$ is the variable in our quadratic, it is unbounded and has asymptotes, making it difficult to visualize solutions. $\sin \theta$ is bounded between $-1$ and $1$, and its graph is a simple, periodic wave, making it much easier to count intersections within a specific domain.

Q: How do I know if I missed a solution at the boundary?

This is a classic JEE trap! Always check the endpoints of your interval. In this problem, the domain is $[-\frac{7\pi}{6}, \frac{4\pi}{3}]$. When you draw the horizontal lines for $\sin \theta = \frac{1}{2}$ and $\sin \theta = -\frac{\sqrt{3}}{2}$, check if the intersection points fall exactly on or within these boundaries. If they do, they must be counted.

Q: Is there a trick to solving the quadratic discriminant?

Yes! When you calculate $D = 16 + 8\sqrt{3}$, it might not look like a perfect square. However, notice that $16 + 8\sqrt{3} = 4(4 + 2\sqrt{3})$. Since $4 + 2\sqrt{3} = (\sqrt{3} + 1)^2$, the whole discriminant is $4(\sqrt{3} + 1)^2$. Recognizing these patterns saves time and prevents errors.

Question

If $	heta \in [-\frac{7\pi}{6}, \frac{4\pi}{3}]$, then the number of solutions of $\sqrt{3} \operatorname{cosec}^2 	heta - 2(\sqrt{3} - 1) \operatorname{cosec} 	heta - 4 = 0$, is equal to

Options:

(A) 6

(B) 8

(C) 10

(D) 7

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The given equation is:
$$\sqrt{3} \operatorname{cosec}^2 	heta - 2(\sqrt{3} - 1) \operatorname{cosec} 	heta - 4 = 0$$

By substituting $x = \operatorname{cosec} 	heta$, the equation transforms into a standard quadratic form:
$$\sqrt{3}x^2 - 2(\sqrt{3}-1)x - 4 = 0$$

This substitution reveals the underlying structure, allowing us to treat the trigonometric expression as a simple algebraic polynomial.

### The Discriminant Dance

We apply the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = \sqrt{3}$, $b = -2(\sqrt{3}-1)$, and $c = -4$.

First, we calculate the discriminant $D = b^2 - 4ac$:
$$D = [-2(\sqrt{3}-1)]^2 - 4(\sqrt{3})(-4)$$
$$D = 4(3 + 1 - 2\sqrt{3}) + 16\sqrt{3}$$
$$D = 16 - 8\sqrt{3} + 16\sqrt{3} = 16 + 8\sqrt{3}$$

We simplify the discriminant by recognizing it as a perfect square:
$$D = 4(4 + 2\sqrt{3}) = 4(\sqrt{3} + 1)^2$$
$$\sqrt{D} = 2(\sqrt{3} + 1)$$

Substituting these values back into the quadratic formula:
$$x = \frac{2(\sqrt{3}-1) \pm 2(\sqrt{3}+1)}{2\sqrt{3}}$$

This yields two distinct roots for $x$:
$$x_1 = \frac{2\sqrt{3} - 2 + 2\sqrt{3} + 2}{2\sqrt{3}} = \frac{4\sqrt{3}}{2\sqrt{3}} = 2$$
$$x_2 = \frac{2\sqrt{3} - 2 - 2\sqrt{3} - 2}{2\sqrt{3}} = \frac{-4}{2\sqrt{3}} = -\frac{2}{\sqrt{3}}$$

### The Graphical Odyssey

Returning to trigonometry, we set $\operatorname{cosec} 	heta = x$, which implies $\sin 	heta = \frac{1}{x}$. Thus, we solve for:
$$\sin 	heta = \frac{1}{2} \quad 	ext{and} \quad \sin 	heta = -\frac{\sqrt{3}}{2}$$

We analyze these within the restricted domain $	heta \in [-\frac{7\pi}{6}, \frac{4\pi}{3}]$.

For $\sin 	heta = \frac{1}{2}$, the solutions are:
$$	heta = \frac{\pi}{6}, \frac{5\pi}{6}, -\frac{7\pi}{6}$$

For $\sin 	heta = -\frac{\sqrt{3}}{2}$, the solutions are:
$$	heta = -\frac{\pi}{3}, -\frac{2\pi}{3}, \frac{4\pi}{3}$$

Summing these individual points, we find a total of **6 solutions**.

Animated Solution for Mathematics - Trigonometry: If $θ \in [- \frac{7 π}{6}, \frac{4 π}{3}]$ , then the number of solutions of $3 cosec^{2} θ - 2 (3 - 1) cosec θ - 4 = 0$ , is equal to

Select Answer:

Visualized Solution

Identify the Quadratic Form

The Quadratic Formula

Calculate $b^{2}$

Calculate $- 4 a c$

Find the Discriminant $D$

Simplify $D$

Solve for $cosec θ$

Convert to $sin θ$

Analyze the Domain

Solutions for $sin θ = \frac{1}{2}$

Solutions for $sin θ = - \frac{3}{2}$

Final Count & Conclusion

The Sigma Insight: General Solution of Trigonometric Equations

Analyzing the Setup

The Discriminant Dance

The Graphical Odyssey

Similar Questions

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Animated Solution for Mathematics - Trigonometry: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If θ∈[−67π​,34π​], then the number of solutions of 3​cosec2θ−2(3​−1)cosecθ−4=0, is equal to

Select Answer:

Visualized Solution

Identify the Quadratic Form

The Quadratic Formula

Calculate .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } b2

Calculate .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } −4ac

Find the Discriminant .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } D

Simplify .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } D​

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

Convert to .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } sinθ

Analyze the Domain

Solutions for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } sinθ=21​

Solutions for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } sinθ=−23​​

Final Count & Conclusion

The Sigma Insight: General Solution of Trigonometric Equations

Analyzing the Setup

The Discriminant Dance

The Graphical Odyssey

Similar Questions

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Animated Solution for Mathematics - Trigonometry: If $θ \in [- \frac{7 π}{6}, \frac{4 π}{3}]$ , then the number of solutions of $3 cosec^{2} θ - 2 (3 - 1) cosec θ - 4 = 0$ , is equal to

Calculate $b^{2}$

Calculate $- 4 a c$

Find the Discriminant $D$

Simplify $D$

Solve for $cosec θ$

Convert to $sin θ$

Solutions for $sin θ = \frac{1}{2}$

Solutions for $sin θ = - \frac{3}{2}$

If $θ \in [- \frac{7 π}{6}, \frac{4 π}{3}]$ , then the number of solutions of $3 cosec^{2} θ - 2 (3 - 1) cosec θ - 4 = 0$ , 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:

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 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 :

Number of solutions of $3 cos 2 θ + 8 cos θ + 33 = 0, θ \in [- 3 π, 2 π]$ is:

The number of solutions of equation $(4 - 3) sin x - 23 cos^{2} x = - \frac{4}{1 + 3}, x \in [- 2 π, \frac{5 π}{2}]$ is

The number of solutions of equation $(4 - 3) sin x - 23 cos^{2} x = - \frac{4}{1 + 3}, x \in [- 2 π, \frac{5 π}{2}]$ is

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