Question

What is the slope of the polar curve `f(theta) = theta - sectheta+thetasin^3theta` at `theta = (7pi)/12`?

Answer 1

`f'((7pi)/12)=-13.8459`

Explanation:

Given:
`f(theta)=theta-sectheta+thetasin^3theta`
Differentiating wrt `theta`
`f'(theta)=1-secthetatantheta+theta(3sin^2theta)costheta+sin^3theta`
`=1-1/costhetasintheta/costheta+sin^3theta+3thetasin^2thetacostheta`
`theta=(7pi)/12=1.8326`

`(7pi)/12=pi-(5pi)/12`
`costheta=cos(pi-(5pi)/12)=-cos((5pi)/12)=-0.2588`
`sintheta=sin(pi-(5pi)/12)=sin((5pi)/12)=0.9659`
Substituting the values in `f'(theta)`

`f'((7pi)/12)=1-1/-0.2588xx0.9659/-0.2588+0.9659^3+3xx1.8326xx0.9659^2(-0.2588)`
`f'((7pi)/12)=-13.8459`