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Write the equation of the line tangent to the graph of r=2cosTheta when Theta =(4pi )/(3)

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Write the equation of the line tangent to the graph of r=2cosTheta  when Theta =(4pi )/(3)

Write the equation of the line tangent to the graph of r=2cosTheta when Theta =(4pi )/(3)

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CarmemElite · Tutor por 8 anos

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To find the equation of the line tangent to the graph of $r=2\cos\Theta$ when $\Theta=\frac{4\pi}{3}$, we need to follow these steps:<br /><br />1. Find the slope of the tangent line at the given point.<br />2. Find the equation of the tangent line using the point-slope form.<br /><br />Step 1: Find the slope of the tangent line at the given point.<br /><br />The slope of the tangent line can be found by taking the derivative of the given equation with respect to $\Theta$ and evaluating it at the given point.<br /><br />Given equation: $r=2\cos\Theta$<br />Taking the derivative with respect to $\Theta$:<br />$\frac{dr}{d\Theta} = -2\sin\Theta$<br /><br />Evaluating the derivative at $\Theta=\frac{4\pi}{3}$:<br />$\frac{dr}{d\Theta}\bigg|_{\Theta=\frac{4\pi}{3}} = -2\sin\left(\frac{4\pi}{3}\right) = -2\left(-\frac{\sqrt{3}}{2}\right) = \sqrt{3}$<br /><br />So, the slope of the tangent line at $\Theta=\frac{4\pi}{3}$ is $\sqrt{3}$.<br /><br />Step 2: Find the equation of the tangent line using the point-slope form.<br /><br />The point-slope form of a line is given by:<br />$y - y_1 = m(x - x_1)$<br /><br />We need to find the coordinates of the point on the graph of $r=2\cos\Theta$ when $\Theta=\frac{4\pi}{3}$.<br /><br />Substituting $\Theta=\frac{4\pi}{3}$ into the given equation:<br />$r = 2\cos\left(\frac{4\pi}{3}\right) = 2\left(-\frac{1}{2}\right) = -1$<br /><br />So, the point on the graph is $\left(-1, \frac{4\pi}{3}\right)$.<br /><br />Now, we can substitute the slope and the coordinates of the point into the point-slope form:<br />$y - \frac{4\pi}{3} = \sqrt{3}(x + 1)$<br /><br />Simplifying the equation:<br />$y = \sqrt{3}x + \sqrt{3} + \frac{4\pi}{3}$<br /><br />Therefore, the equation of the line tangent to the graph of $r=2\cos\Theta$ when $\Theta=\frac{4\pi}{3}$ is:<br />$y = \sqrt{3}x + \sqrt{3} + \frac{4\pi}{3}$
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