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
A line passing through the point $P(\sqrt{5}, \sqrt{5})$ intersects the ellipse $\frac{x^2}{36} + \frac{y^2}{25} = 1$ at A and B such that $(PA).(PB)$ is maximum. Then $5(PA^2 + PB^2)$ is equal to :
- If $\theta \in [ - 2\pi, 2\pi]$, then the number of solutions of $2\sqrt{2}\cos^2\theta+(2-\sqrt{6}) \cos\theta-\sqrt{3}=0$, is equal to:
- The sum of the infinite series $\cot^{-1} \left(\frac{7}{4}\right) + \cot^{-1} \left(\frac{19}{4}\right) + \cot^{-1} \left(\frac{39}{4}\right) + \cot^{-1} \left(\frac{67}{4}\right) + \dots$ is:
- The number of solutions of the equation $(4-\sqrt{3}) \sin x - 2\sqrt{3} \cos^2 x = -\frac{4}{1+\sqrt{3}}, x \in [-2\pi, \frac{5\pi}{2}]$ is
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 :-
The number of solutions of the equation $\cos 2\theta \cos \frac{\theta}{2} + \cos \frac{5\theta}{2} = 2\cos^3 \frac{5\theta}{2}$ in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$is :
The number of solutions of $sin3x = cos2x$, in the interval $\left[ \frac{\pi}{2}, \pi \right]$ is :-
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 ______________.
- 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
Important Questions from Trigonometry
A line passing through the point $P(\sqrt{5}, \sqrt{5})$ intersects the ellipse $\frac{x^2}{36} + \frac{y^2}{25} = 1$ at A and B such that $(PA).(PB)$ is maximum. Then $5(PA^2 + PB^2)$ is equal to :
- If $\theta \in [ - 2\pi, 2\pi]$, then the number of solutions of $2\sqrt{2}\cos^2\theta+(2-\sqrt{6}) \cos\theta-\sqrt{3}=0$, is equal to:
- The sum of the infinite series $\cot^{-1} \left(\frac{7}{4}\right) + \cot^{-1} \left(\frac{19}{4}\right) + \cot^{-1} \left(\frac{39}{4}\right) + \cot^{-1} \left(\frac{67}{4}\right) + \dots$ is:
- The number of solutions of the equation $(4-\sqrt{3}) \sin x - 2\sqrt{3} \cos^2 x = -\frac{4}{1+\sqrt{3}}, x \in [-2\pi, \frac{5\pi}{2}]$ is
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 :-
