Solution
y=(1+2cosθ)⋅sinθ
Differentiate with respect to \thetaθ by using the product rule
\dfrac{dy}{d\theta}=\left(0-2\sin\theta\right)\cdot\sin\theta+\left(1+2\cos\theta\right)\cdot\left(\cos\theta\right)dθdy=(0−2sinθ)⋅sinθ+(1+2cosθ)⋅(cosθ)
Substitute \theta=\pi/3θ=π/3 (which means \sin\theta=\sqrt{3}/2sinθ=3/2 and \cos\theta=1/2cosθ=1/2)
\dfrac{dy}{d\theta}\bigg|_{\theta=\pi/3}=\left(0-2\cdot\dfrac{\sqrt{3}}{2}\right)\cdot\dfrac{\sqrt{3}}{2}+\left(1+2\cdot\dfrac{1}{2}\right)\cdot\left(\dfrac{1}{2}\right)dθdy∣∣θ=π/3=(0−2⋅23)⋅23+(1+2⋅21)⋅(21)
\dfrac{dy}{d\theta}\bigg|_{\theta=\pi/3}=\left(-\sqrt{3}\right)\cdot\dfrac{\sqrt{3}}{2}+\left(1+1\right)\cdot\left(\dfrac{1}{2}\right)dθdy∣∣θ=π/3=(−3)⋅23+(1+1)⋅(21)
\dfrac{dy}{d\theta}\bigg|_{\theta=\pi/3}=-\dfrac{3}{2}+1=-\dfrac{3}{2}+\dfrac{2}{2}=-\dfrac{1}{2}dθdy∣∣θ=π/3=−23+1=−23+22=−21
Using the chain rule, we can write
\dfrac{dy}{dx}\bigg|_{\theta=\pi/3}=\dfrac{dy/d\theta}{dx/d\theta}\bigg|_{\theta=\pi/3}=\dfrac{-1/2}{-3\sqrt{3}/2}=\dfrac{1}{3\sqrt{3}}dxdy∣∣θ=π/3=dx/dθdy/dθ∣∣θ=π/3=−33/2−1/2=331
Slope of the tangent is \dfrac{1}{3\sqrt{3}}331