Trigonometric Function Simplification
First, simplify the trigonometric terms in the function $f(\theta)$ using angle properties and periodicity:
- $\sin\left(\frac{7\pi}{2} - \theta\right) = \sin\left(3\pi + \frac{\pi}{2} - \theta\right) = \sin\left(\pi + \left(\frac{\pi}{2} - \theta\right)\right) = -\sin\left(\frac{\pi}{2} - \theta\right) = -\cos(\theta)$
- $\sin(11\pi + \theta) = \sin(\pi + \theta) = -\sin(\theta)$
- $\sin\left(\frac{3\pi}{2} - \theta\right) = -\cos(\theta)$
- $\sin(9\pi - \theta) = \sin(\pi - \theta) = \sin(\theta)$
Substitute these back into the function:
$f(\theta) = 4\left((-\cos\theta)^4 + (-\sin\theta)^4\right) - 2\left((-\cos\theta)^6 + (\sin\theta)^6\right)$
$f(\theta) = 4\left(\cos^4\theta + \sin^4\theta\right) - 2\left(\cos^6\theta + \sin^6\theta\right)$
Use the identities:
- $\cos^4\theta + \sin^4\theta = 1 - 2\sin^2\theta\cos^2\theta$
- $\cos^6\theta + \sin^6\theta = 1 - 3\sin^2\theta\cos^2\theta$
Substitute these identities:
$f(\theta) = 4(1 - 2\sin^2\theta\cos^2\theta) - 2(1 - 3\sin^2\theta\cos^2\theta)$
$f(\theta) = 4 - 8\sin^2\theta\cos^2\theta - 2 + 6\sin^2\theta\cos^2\theta$
$f(\theta) = 2 - 2\sin^2\theta\cos^2\theta$
Using the double angle identity $\sin(2\theta) = 2\sin\theta\cos\theta$, we get $\sin^2\theta\cos^2\theta = \frac{\sin^2(2\theta)}{4}$:
$f(\theta) = 2 - 2\left(\frac{\sin^2(2\theta)}{4}\right) = 2 - \frac{1}{2}\sin^2(2\theta)$
Finding Extrema $(\alpha, \beta)$
The simplified function is $f(\theta) = 2 - \frac{1}{2}\sin^2(2\theta)$.
We know that the range of $\sin(x)$ is $[-1, 1]$, so the range of $\sin^2(x)$ is $[0, 1]$.
Therefore, the range of $\sin^2(2\theta)$ is $[0, 1]$.
To find the maximum value ($\alpha$), we minimize $\sin^2(2\theta)$:
$\alpha = \max(f(\theta)) = 2 - \frac{1}{2}(0) = 2$
To find the minimum value ($\beta$), we maximize $\sin^2(2\theta)$:
$\beta = \min(f(\theta)) = 2 - \frac{1}{2}(1) = 2 - \frac{1}{2} = \frac{3}{2}$
Final Calculation
The question asks for the value of $\alpha + 2\beta$.
Substitute the found values of $\alpha$ and $\beta$:
$\alpha + 2\beta = 2 + 2\left(\frac{3}{2}\right)$
$\alpha + 2\beta = 2 + 3 = 5$
Similar Questions
Let $\cos(\alpha + \beta) = -\frac{1}{10}$ and $\sin(\alpha - \beta) = \frac{3}{8}$, where $0 < \alpha < \frac{\pi}{3}$ and $0 < \beta < \frac{\pi}{4}$. If $\tan 2\alpha = \frac{3(1 - r\sqrt{5})}{\sqrt{11}(s + \sqrt{5})}, r, s \in \mathbb{N}$, then $r+s$ is equal to ________.
- The number of elements in the set $\{x \in [0, 180^\circ] : \tan(x + 100^\circ) = \tan(x + 50^\circ) \tan x \tan(x - 50^\circ)\}$ is _______.
- If $\frac{\pi}{4} + \sum_{p=1}^{11} \tan^{-1} \left( \frac{2^{p-1}}{1 + 2^{2p-1}} \right) = \alpha$, then $\tan \alpha$ is equal to _________.
Let $\vec{a_k} = (\tan \theta_k) \hat{i} + \hat{j}$ and $\vec{b_k} = \hat{i} - (\cot \theta_k) \hat{j}$, where $\theta_k = \frac{2^{k - 1}\pi}{2^n + 1}$, for some $n \in \mathbb{N}, n > 5$. Then the value of $\frac{\sum_{k=1}^n |\vec{a_k}|^2}{\sum_{k=1}^n |\vec{b_k}|^2}$ is _____.
- If $k = \tan\left(\frac{\pi}{4} + \frac{1}{2}\cos^{-1}\left(\frac{2}{3}\right)\right) + \tan\left(\frac{1}{2}\sin^{-1}\left(\frac{2}{3}\right)\right)$, then the number of solutions of the equation $\sin^{-1}(kx-1) = \sin^{-1}x - \cos^{-1}x$ is ________
- Let the maximum value of $(\sin^{-1} x)^2 + (\cos^{-1} x)^2$ for $x \in \left[-\frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}}\right]$ be $\frac{m}{n} \pi^2$, where $\gcd(m, n) = 1$. Then $m + n$ is equal to __________.
- The angle of elevation of the top $P$ of a vertical tower for a person standing at a point $A$ on the horizontal ground due north of the tower is $45^\circ$. Another person $B$ is standing $50$ m west of $A$ on the ground. If the angle of elevation of $P$ at $B$ is $30^\circ$ then the height (in meters) of the tower is
- Let $S = \{\theta \in [0,2\pi] - (\frac{\pi}{2}, \frac{3\pi}{2}) \mid \sin^{-1}(\sin\theta) + \cos^{-1}(\cos\theta) + \tan^{-1}(\tan\theta) = \frac{4\pi}{5}\}$ Then $\frac{30}{\pi} \sum_{\theta \in S} \theta$ is equal to ________
Consider the following two statements :-
Statement p :
The value of $sin120^\circ$ can be derived by taking $\theta = 240^\circ$ in the equation $2sin \frac{\theta}{2} = \sqrt{1+sin \theta} - \sqrt{1-sin \theta}$
Statement q :
The angles A, B, C and D of any quadrilateral ABCD satisfy the equation $cos \left( \frac{1}{2}(A+C) \right) + cos \left( \frac{1}{2}(B+D) \right) = 0$
Then the truth values of p and q are respectively :-
- A tower $T_1$ of height 60 m is located exactly opposite to a tower $T_2$ of height 80 m on a straight road. From the top of $T_1$, if the angle of depression of the foot of $T_2$ is twice the angle of elevaion of the top of $T_2$, then the width (in m) of the road between the feet of the towers $T_1$ and $T_2$ is :-
Important Questions from Trigonometry
Let $\cos(\alpha + \beta) = -\frac{1}{10}$ and $\sin(\alpha - \beta) = \frac{3}{8}$, where $0 < \alpha < \frac{\pi}{3}$ and $0 < \beta < \frac{\pi}{4}$. If $\tan 2\alpha = \frac{3(1 - r\sqrt{5})}{\sqrt{11}(s + \sqrt{5})}, r, s \in \mathbb{N}$, then $r+s$ is equal to ________.
- The number of elements in the set $\{x \in [0, 180^\circ] : \tan(x + 100^\circ) = \tan(x + 50^\circ) \tan x \tan(x - 50^\circ)\}$ is _______.
- If $\frac{\pi}{4} + \sum_{p=1}^{11} \tan^{-1} \left( \frac{2^{p-1}}{1 + 2^{2p-1}} \right) = \alpha$, then $\tan \alpha$ is equal to _________.
Let $\vec{a_k} = (\tan \theta_k) \hat{i} + \hat{j}$ and $\vec{b_k} = \hat{i} - (\cot \theta_k) \hat{j}$, where $\theta_k = \frac{2^{k - 1}\pi}{2^n + 1}$, for some $n \in \mathbb{N}, n > 5$. Then the value of $\frac{\sum_{k=1}^n |\vec{a_k}|^2}{\sum_{k=1}^n |\vec{b_k}|^2}$ is _____.
- If $k = \tan\left(\frac{\pi}{4} + \frac{1}{2}\cos^{-1}\left(\frac{2}{3}\right)\right) + \tan\left(\frac{1}{2}\sin^{-1}\left(\frac{2}{3}\right)\right)$, then the number of solutions of the equation $\sin^{-1}(kx-1) = \sin^{-1}x - \cos^{-1}x$ is ________
