If θ1 and θ2 be respectively the smallest and the largest values of θ in (0,2π) - {π} which satisfy

Q: Why do we reject $\csc \theta = \frac{1}{2}$? Isn't it a valid root of the quadratic?

Excellent question! While it is a root of the algebraic quadratic, it is not a valid value for the cosecant function. Remember, $\csc \theta = \frac{1}{\sin \theta}$. Since the range of $\sin \theta$ is $[-1, 1]$, the range of $\csc \theta$ must be $(-\infty, -1] \cup [1, \infty)$. Thus, $\frac{1}{2}$ is impossible.

Q: How do I know which values of $\theta$ are $\theta_1$ and $\theta_2$?

Visualize the unit circle. We found $\sin \theta = \frac{1}{2}$. This occurs in the first quadrant ($\frac{\pi}{6}$) and the second quadrant ($\frac{5\pi}{6}$). Since the problem asks for the smallest and largest values in $(0, 2\pi) - \{\pi\}$, these are your boundaries.

Q: Can I use integration by parts for $\cos^2 3\theta$?

You could, but it is like using a sledgehammer to crack a nut! The power reduction identity is the elegant, standard path. It simplifies the integrand immediately, saving you time and reducing the chance of calculation errors.

Question

If $	heta_1$ and $	heta_2$ be respectively the smallest and the largest values of $	heta$ in $(0, 2\pi) - \{\pi\}$ which satisfy the equation, $2 \cot^2 	heta - \frac{5}{\sin 	heta} + 4 = 0$, then $\int_{	heta_1}^{	heta_2} \cos^2 3	heta d	heta$ is equal to :

Options:

(A) $\frac{2\pi}{3}$

(B) $\frac{\pi}{3}$

(C) $\frac{\pi}{3} + \frac{1}{6}$

(D) $\frac{\pi}{9}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We begin with the trigonometric equation:
$$2 \cot^2 	heta - \frac{5}{\sin 	heta} + 4 = 0$$

To solve this, we must unify the trigonometric functions. Using the identity $\cot^2 	heta = \csc^2 	heta - 1$ and the definition $\csc 	heta = \frac{1}{\sin 	heta}$, we substitute these into the equation:
$$2(\csc^2 	heta - 1) - 5 \csc 	heta + 4 = 0$$

This simplifies to a standard quadratic equation in terms of $\csc 	heta$:
$$2 \csc^2 	heta - 5 \csc 	heta + 2 = 0$$

### The Trap: Respecting the Domain

Next, we factorize the quadratic expression:
$$(2 \csc 	heta - 1)(\csc 	heta - 2) = 0$$

This yields two potential solutions: $\csc 	heta = \frac{1}{2}$ or $\csc 	heta = 2$.

However, we must respect the domain of the cosecant function, which is $(-\infty, -1] \cup [1, \infty)$. Since $\csc 	heta = \frac{1}{2}$ falls outside this range, we reject it.

We are left with $\csc 	heta = 2$, which implies $\sin 	heta = \frac{1}{2}$. Within the interval $(0, 2\pi)$, the solutions are:
$$	heta_1 = \frac{\pi}{6}, \quad 	heta_2 = \frac{5\pi}{6}$$

### The Calculus Waltz: Integration with Elegance

With our limits secured, we evaluate the integral:
$$I = \int_{\pi/6}^{5\pi/6} \cos^2 3	heta \, d	heta$$

Using the power reduction identity $\cos^2 A = \frac{1 + \cos 2A}{2}$, we set $A = 3	heta$ to rewrite the integrand:
$$I = \frac{1}{2} \int_{\pi/6}^{5\pi/6} (1 + \cos 6	heta) \, d	heta$$

Performing the integration term by term, we obtain:
$$I = \frac{1}{2} \left[ 	heta + \frac{\sin 6	heta}{6} ight]_{\pi/6}^{5\pi/6}$$

Evaluating at the boundaries, the sine terms vanish because $\sin(5\pi) = 0$ and $\sin(\pi) = 0$. This leaves us with:
$$I = \frac{1}{2} \left( \frac{5\pi}{6} - \frac{\pi}{6} ight) = \frac{1}{2} \left( \frac{4\pi}{6} ight)$$

The final result is:
$$\mathbf{I = \frac{\pi}{3}}$$

Animated Solution for Mathematics - Trigonometry: 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 :

Select Answer:

Visualized Solution

The Given Equation

Using Trigonometric Identity

Substituting the Identity

Forming the Quadratic Equation

Solving the Quadratic

Analyzing the Solutions

Finding $θ_{1}$ and $θ_{2}$

Setting up the Integral

Applying Power Reduction

Performing Integration

Evaluating the Limits

Final Conclusion

The Sigma Insight: General Solution of Trigonometric Equations

Analyzing the Setup

The Trap

The Calculus Waltz

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