If $y=\cos\left(\frac{\pi}{3}+\cos^{-1}\frac{x}{2}\right)$, then $(x-y)^2+3y^2$ is equal to ____________.
We are given the expression $y=\cos\left(\frac{\pi}{3}+\cos^{-1}\frac{x}{2}\right)$ and asked to find the value of $(x-y)^2+3y^2$.
Simplifying the Cosine Expression
Let $\theta = \cos^{-1}\frac{x}{2}$. This implies $\cos\theta = \frac{x}{2}$. We know that for $\theta$ in the range of $\cos^{-1}$, which is $[0, \pi]$, $\sin\theta \ge 0$. Therefore, $\sin\theta = \sqrt{1-\cos^2\theta} = \sqrt{1-\left(\frac{x}{2}\right)^2} = \frac{\sqrt{4-x^2}}{2}$.
Now substitute this into the expression for $y$ using the cosine addition formula $\cos(A+B) = \cos A \cos B - \sin A \sin B$:
$y = \cos\left(\frac{\pi}{3}+\theta\right)$ $y = \cos\frac{\pi}{3}\cos\theta - \sin\frac{\pi}{3}\sin\theta$Using the values $\cos\frac{\pi}{3} = \frac{1}{2}$ and $\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}$, we get:
$y = \left(\frac{1}{2}\right)\left(\frac{x}{2}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{4-x^2}}{2}\right)$ $y = \frac{x}{4} - \frac{\sqrt{3}\sqrt{4-x^2}}{4}$ $y = \frac{x - \sqrt{12-3x^2}}{4}$Evaluating the Target Expression
Multiply by 4:
$4y = x - \sqrt{12-3x^2}$Rearrange to isolate the square root term:
$4y - x = -\sqrt{12-3x^2}$Square both sides:
$(4y - x)^2 = \left(-\sqrt{12-3x^2}\right)^2$ $16y^2 - 8xy + x^2 = 12 - 3x^2$Move all terms to one side:
$16y^2 - 8xy + x^2 + 3x^2 = 12$ $16y^2 - 8xy + 4x^2 = 12$Divide the entire equation by 4:
$4y^2 - 2xy + x^2 = 3$Now, let's expand the expression we need to evaluate:
$(x-y)^2 + 3y^2 = (x^2 - 2xy + y^2) + 3y^2$ $(x-y)^2 + 3y^2 = x^2 - 2xy + 4y^2$Comparing this with the equation derived ($x^2 - 2xy + 4y^2 = 3$), we see that the value of the expression is 3.
Final Result
Therefore, $(x-y)^2+3y^2 = 3$.
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 ________
Let A(4, -2), B(1, 1) and C(9, -3) be the vertices of a triangle ABC. Then the maximum area of the parallelogram AFDE, formed with vertices D, E and F on the sides BC, CA and AB of the triangle ABC respectively, is ______________.
- If $\text{A} = \frac{\sin 3^\circ}{\cos 9^\circ} + \frac{\sin 9^\circ}{\cos 27^\circ} + \frac{\sin 27^\circ}{\cos 81^\circ}$ and $\text{B} = \tan 81^\circ - \tan 3^\circ$, then $\frac{\text{B}}{\text{A}}$ is equal to _____.
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 ________
