Word Count : 386

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The parametric equations of the cycloid are: [ x = \alpha(\theta - \sin \theta), \quad y = \alpha(1 - \cos \theta) ] The length of a curve given parametrically is [ L = \int_0^{\theta_1} \sqrt{\left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2}, d\theta ] First, compute derivatives: [ \frac{dx}{d\theta} = \alpha(1 - \cos \theta), \quad \frac{dy}{d\theta} = \alpha \sin \theta ] Then [ \left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2 = \alpha^2 (1 - \cos \theta)^2 + \alpha^2 (\sin \theta)^2 = \alpha^2 [(1 - \cos \theta)^2 + \sin^2 \theta] ] Simplify using (\sin^2 \theta + (1 - \cos \theta)^2): [ (1 - \cos \theta)^2 + ______ _____ ______ _____ ______ ___ ____ _______ _______ ______.
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