QuestionJune 7, 2025

(4 pts) 8) Find the equation of the tangent line to the parametric curve ) x=tcost y=tsint when t=(pi )/(2)

(4 pts) 8) Find the equation of the tangent line to the parametric curve ) x=tcost y=tsint when t=(pi )/(2)
(4 pts) 8) Find the equation of the tangent line to the parametric curve ) x=tcost y=tsint when t=(pi )/(2)

Solution
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Answer

y = -\frac{2}{\pi}x + \frac{\pi}{2} Explanation 1. Find derivatives Differentiate x = t \cos t and y = t \sin t with respect to t. - \frac{dx}{dt} = \cos t - t \sin t - \frac{dy}{dt} = \sin t + t \cos t 2. Calculate slope at t = \frac{\pi}{2} Use the formula for slope of tangent line: \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}. Evaluate at t = \frac{\pi}{2}. - \frac{dx}{dt} = \cos\left(\frac{\pi}{2}\right) - \frac{\pi}{2} \sin\left(\frac{\pi}{2}\right) = -\frac{\pi}{2} - \frac{dy}{dt} = \sin\left(\frac{\pi}{2}\right) + \frac{\pi}{2} \cos\left(\frac{\pi}{2}\right) = 1 - Slope m = \frac{1}{-\frac{\pi}{2}} = -\frac{2}{\pi} 3. Find point on curve at t = \frac{\pi}{2} Substitute t = \frac{\pi}{2} into parametric equations to find (x, y). - x = \frac{\pi}{2} \cos\left(\frac{\pi}{2}\right) = 0 - y = \frac{\pi}{2} \sin\left(\frac{\pi}{2}\right) = \frac{\pi}{2} 4. Write equation of tangent line Use point-slope form: y - y_1 = m(x - x_1). - y - \frac{\pi}{2} = -\frac{2}{\pi}(x - 0) - Simplify: y = -\frac{2}{\pi}x + \frac{\pi}{2}

Explanation

1. Find derivatives<br /> Differentiate $x = t \cos t$ and $y = t \sin t$ with respect to $t$. <br />- $\frac{dx}{dt} = \cos t - t \sin t$<br />- $\frac{dy}{dt} = \sin t + t \cos t$<br /><br />2. Calculate slope at $t = \frac{\pi}{2}$<br /> Use the formula for slope of tangent line: $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$. Evaluate at $t = \frac{\pi}{2}$.<br />- $\frac{dx}{dt} = \cos\left(\frac{\pi}{2}\right) - \frac{\pi}{2} \sin\left(\frac{\pi}{2}\right) = -\frac{\pi}{2}$<br />- $\frac{dy}{dt} = \sin\left(\frac{\pi}{2}\right) + \frac{\pi}{2} \cos\left(\frac{\pi}{2}\right) = 1$<br />- Slope $m = \frac{1}{-\frac{\pi}{2}} = -\frac{2}{\pi}$<br /><br />3. Find point on curve at $t = \frac{\pi}{2}$<br /> Substitute $t = \frac{\pi}{2}$ into parametric equations to find $(x, y)$.<br />- $x = \frac{\pi}{2} \cos\left(\frac{\pi}{2}\right) = 0$<br />- $y = \frac{\pi}{2} \sin\left(\frac{\pi}{2}\right) = \frac{\pi}{2}$<br /><br />4. Write equation of tangent line<br /> Use point-slope form: $y - y_1 = m(x - x_1)$.<br />- $y - \frac{\pi}{2} = -\frac{2}{\pi}(x - 0)$<br />- Simplify: $y = -\frac{2}{\pi}x + \frac{\pi}{2}$
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