For the polar functions of r = V3 Cos 20, calculate the radial lines, graph and calculate the area of the region inside the curve



Answer to a math question For the polar functions of r = V3 Cos 20, calculate the radial lines, graph and calculate the area of the region inside the curve

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To calculate the radial lines for the polar function r = \sqrt{3} \cos(2\theta) , we need to set r to zero to determine where the function crosses the origin.

1. Set r = 0 :
\sqrt{3} \cos(2\theta) = 0

2. Solve for \theta :
\cos(2\theta) = 0
2\theta = \frac{\pi}{2}, \ \frac{3\pi}{2}
\theta = \frac{\pi}{4}, \ \frac{3\pi}{4}

So, the radial lines are at \frac{\pi}{4} and \frac{3\pi}{4} .

Now, to plot the graph of the polar function r = \sqrt{3} \cos(2\theta) , we have to sketch the curve based on the equation.

Next, to calculate the area of the region inside the curve, we'll integrate the polar function from \theta = \frac{\pi}{4} to \theta = \frac{3\pi}{4} and then double the result, because the region is symmetric about the x-axis.

Now, calculate the area:

A = 2\int_{\pi/4}^{3\pi/4} \frac{1}{2}(\sqrt{3}\cos(2\theta))^2 d\theta
= 2\int_{\pi/4}^{3\pi/4} \frac{3}{2} \cos^2(2\theta) d\theta
= 2\int_{\pi/4}^{3\pi/4} \frac{3}{4}(1 + \cos(4\theta)) d\theta
= 2\left[\frac{3}{4}(\theta + \frac{1}{4}\sin(4\theta))\right]_{\pi/4}^{3\pi/4}
= 2\left[\frac{3}{4}\left(\frac{3\pi}{4} - \frac{\pi}{4} + \frac{1}{4}\sin(3\pi) - \frac{1}{4}\sin(\pi)\right)\right]
= 2\left[\frac{3}{4}\left(\frac{\pi}{2} + \frac{1}{4}(0) - \frac{1}{4}(0)\right)\right]
= 2\left[\frac{3}{4}\cdot\frac{\pi}{2}\right]
= \frac{3\pi}{2}

Therefore, the area of the region inside the curve is \frac{3\pi}{2} .

\boxed{A = \frac{3\pi}{2}}

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