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 :-
Statement p Analysis
We need to verify if $2\sin \frac{\theta}{2} = \sqrt{1+\sin \theta} - \sqrt{1-\sin \theta}$ holds true for $\theta = 240^\circ$.
Let $\theta = 240^\circ$, then $\frac{\theta}{2} = 120^\circ$. The left side (LHS) is $2\sin(120^\circ) = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3}$.
For the right side (RHS), we use the identities: $1 + \sin \theta = (\sin(\theta/2) + \cos(\theta/2))^2$ $1 - \sin \theta = (\cos(\theta/2) - \sin(\theta/2))^2$ So, $\sqrt{1+\sin \theta} = |\sin(\theta/2) + \cos(\theta/2)|$ And $\sqrt{1-\sin \theta} = |\cos(\theta/2) - \sin(\theta/2)|$
Substitute $\theta/2 = 120^\circ$: $\sin(120^\circ) = \frac{\sqrt{3}}{2}$ $\cos(120^\circ) = -\frac{1}{2}$
Calculate the terms inside the absolute values: $\sin(120^\circ) + \cos(120^\circ) = \frac{\sqrt{3}}{2} - \frac{1}{2} = \frac{\sqrt{3}-1}{2}$ (Positive) $\cos(120^\circ) - \sin(120^\circ) = -\frac{1}{2} - \frac{\sqrt{3}}{2} = -\frac{1+\sqrt{3}}{2}$ (Negative)
Evaluate the absolute values: $|\sin(120^\circ) + \cos(120^\circ)| = \frac{\sqrt{3}-1}{2}$ $|\cos(120^\circ) - \sin(120^\circ)| = |-\frac{1+\sqrt{3}}{2}| = \frac{1+\sqrt{3}}{2}$
RHS = $|\sin(120^\circ) + \cos(120^\circ)| - |\cos(120^\circ) - \sin(120^\circ)| = \frac{\sqrt{3}-1}{2} - \frac{1+\sqrt{3}}{2} = \frac{\sqrt{3}-1 - 1 - \sqrt{3}}{2} = \frac{-2}{2} = -1$.
Since LHS ($\sqrt{3}$) is not equal to RHS (-1), Statement p is False.
Statement q Analysis
Let A, B, C, D be the angles of a quadrilateral. The sum of the angles is $A+B+C+D = 360^\circ$. We need to check if $\cos \left( \frac{1}{2}(A+C) \right) + \cos \left( \frac{1}{2}(B+D) \right) = 0$.
From the sum of angles, $A+C = 360^\circ - (B+D)$. Dividing by 2, we get $\frac{1}{2}(A+C) = \frac{1}{2}(360^\circ - (B+D)) = 180^\circ - \frac{1}{2}(B+D)$.
Substitute this into the equation: $\cos \left( 180^\circ - \frac{1}{2}(B+D) \right) + \cos \left( \frac{1}{2}(B+D) \right)$
Using the trigonometric identity $\cos(180^\circ - x) = -\cos x$, where $x = \frac{1}{2}(B+D)$: $-\cos \left( \frac{1}{2}(B+D) \right) + \cos \left( \frac{1}{2}(B+D) \right) = 0$.
The equation $0 = 0$ is true. Therefore, Statement q is True.
Conclusion
Statement p is False (F) and Statement q is True (T). The truth values are (F, T).
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 ________
- 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 :-
The number of solutions of $sin3x = cos2x$, in the interval $\left[ \frac{\pi}{2}, \pi \right]$ 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 ________
