Question

Find the slope of the tangent line to the given polar curve at the point specified by the value of \({\rm{\theta }}\)

\({\rm{r = 2 - sin\theta ,\theta = }}\frac{{\rm{\pi }}}{{\rm{3}}}\)

Short answer

The slope of tangent line is\(\frac{{{\rm{2 - }}\sqrt {\rm{3}} }}{{{\rm{1 - 2}}\sqrt {\rm{3}} }}\).

Step-by-step solution

Expand slope of tangent line.

\({\rm{r = 2 - sin\theta }}\;\;\;{\rm{\theta = }}\frac{{\rm{\pi }}}{{\rm{3}}}\)

The slope of the tangent line

\(\begin{aligned}{c}{\rm{ = }}\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\\{\rm{ = }}\frac{{\frac{{{\rm{dr}}}}{{{\rm{d\theta }}}}{\rm{sin\theta + rcos\theta }}}}{{\frac{{{\rm{dr}}}}{{{\rm{d\theta }}}}{\rm{cos\theta - rsin\theta }}}}\\\frac{{{\rm{dy}}}}{{{\rm{dx}}}}{\rm{ = }}\frac{{{\rm{ - cos\theta sin\theta + (2 - sin\theta )cos\theta }}}}{{{\rm{ - cos\theta cos\theta - (2 - sin\theta )sin\theta }}}}\end{aligned}\)

Find the slope of tangent at point\({\rm{\theta  = }}\frac{{\rm{\pi }}}{{\rm{3}}}\).

The slope of the tangent at the point where\({\rm{\theta = }}\frac{{\rm{\pi }}}{{\rm{3}}}\)

\(\begin{aligned}{c}\frac{{{\rm{dy}}}}{{{\rm{dx}}}}{\rm{ = }}\frac{{{\rm{ - cos(\pi /3)sin(\pi /3) + (2 - sin(\pi /3))cos(\pi /3)}}}}{{{\rm{ - cos(\pi /3)cos(\pi /3) - (2 - sin(\pi /3))sin(\pi /3)}}}}\\{\rm{ = }}\frac{{{\rm{2 - }}\sqrt {\rm{3}} }}{{{\rm{1 - 2}}\sqrt {\rm{3}} }}\end{aligned}\)

Therefore, the slope of the tangent line is\(\frac{{{\rm{2 - }}\sqrt {\rm{3}} }}{{{\rm{1 - 2}}\sqrt {\rm{3}} }}\).