# What is the equation of the tangent line to the polar curve  f(theta)=2thetasin(theta)+thetacot^2(4theta)  at theta = pi/3?

Aug 7, 2018

Equation of tangent is $y = - 3.05 x + 5.36$

#### Explanation:

f(theta) = 2 theta sin theta + theta cot^2 (4 theta); theta= pi/3

$f \left(\frac{\pi}{3}\right) = 2 \cdot \frac{\pi}{3} \cdot \sin \left(\frac{\pi}{3}\right) + \frac{\pi}{3} \cdot {\cot}^{2} \left(4 \cdot \frac{\pi}{3}\right)$

$f \left(\frac{\pi}{3}\right) \approx 2.16 \therefore$ Point is $\left(\frac{\pi}{3} , 2.16\right)$

f(theta) = 2 theta sin theta + theta cot^2 (4 theta);

${f}^{'} \left(\theta\right) = 2 \sin \theta + 2 \theta \cos \theta + {\cot}^{2} \left(4 \theta\right) - 8 \cot \left(4 \theta\right) {\csc}^{2} \left(4 \theta\right)$

f^'(pi/3) = 2*sin( (pi)/3) +2*pi/3*cos((pi)/3)+cot^2 (4*(pi)/3))-8 cot (4*(pi)/3)csc^2(4* (pi)/3)

${f}^{'} \left(\frac{\pi}{3}\right) \approx - 3.05 \therefore$ slope at point $\left(1.05 , 2.16\right)$ is $- 3.05$

Equation of tangent is $y - 2.16 = - 3.05 \left(x - 1.05\right)$ or

$y = - 3.05 x + 3.2 + 2.16 \mathmr{and} y = - 3.05 x + 5.36$ [Ans]