Question

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. x = cos(θ) + sin(4θ) y = sin(θ) + cos(4θ) θ = 0

Answer 1

The line tangent to the curve at $(x(0),y(0))$ has slope

$\dfrac{\mathrm dy}{\mathrm dx}\bigg|_{\theta=0}=\dfrac{\mathrm dy}{\mathrm d\theta}\dfrac{\mathrm d\theta}{\mathrm dx}\bigg|_{\theta=0}=\dfrac{\cos\theta-4\sin4\theta}{-\sin\theta+4\cos4\theta}\bigg|_{\theta=0}=\dfrac14$

Then the tangent line has equation

$y-y(0)=\dfrac14(x-x(0))$

$y-1=\dfrac14(x-1)$

$\boxed{y=\dfrac{x+3}4}$