Question

Find the slope of the tangent line to the given polar curve at the point specified by the value of \({\rm{\theta }}\). \({\rm{r = 2sin\theta ,}}\;\;\;{\rm{\theta = }}\pi {\rm{/6}}\)

Short answer

The value of \({\rm{\theta }}\)is \(\sqrt {\rm{3}} \).

Step-by-step solution

Finding values of given equations.

\(\begin{aligned}{l}{\rm{r = 2sin\theta }}\\{\rm{\theta = }}\pi {\rm{/6}}\end{aligned}\)

First find the value,

\(\frac{{{\rm{dr}}}}{{{\rm{d\theta }}}}{\rm{ = 2cos\theta }}\)

\({\rm{r}}\)into,

\(\begin{aligned}{c}\frac{{{\rm{dy}}}}{{{\rm{dx}}}}{\rm{ = }}\frac{{\frac{{{\rm{dr}}}}{{{\rm{d\theta }}}}{\rm{sin\theta + rcos\theta }}}}{{\frac{{{\rm{dr}}}}{{{\rm{d\theta }}}}{\rm{cos\theta - rsin\theta }}}}\\{\rm{ = }}\frac{{{\rm{2cos\theta sin\theta + 2sin\theta cos\theta }}}}{{{\rm{2cos\theta cos\theta - 2sin\theta sin\theta }}}}\\{\rm{ = }}\frac{{{\rm{2sin2\theta }}}}{{{\rm{2co}}{{\rm{s}}^{\rm{2}}}{\rm{\theta - 2si}}{{\rm{n}}^{\rm{2}}}{\rm{\theta }}}}\\{\rm{ = }}\frac{{{\rm{sin2\theta }}}}{{{\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{\theta - si}}{{\rm{n}}^{\rm{2}}}{\rm{\theta }}}}\end{aligned}\)

Solve the equation.

The value of\({\rm{\theta = }}\pi {\rm{/6}}\)

\(\begin{aligned}{c}\frac{{{\rm{dy}}}}{{{\rm{dx}}}}{\rm{ = }}\frac{{{\rm{sin}}\frac{\pi }{{\rm{3}}}}}{{{\rm{co}}{{\rm{s}}^{\rm{2}}}\frac{\pi }{{\rm{6}}}{\rm{ - si}}{{\rm{n}}^{\rm{2}}}\frac{\pi }{{\rm{6}}}}}\\{\rm{ = }}\frac{{\frac{{\sqrt {\rm{3}} }}{{\rm{2}}}}}{{\frac{{\rm{3}}}{{\rm{4}}}{\rm{ - }}\frac{{\rm{1}}}{{\rm{4}}}}}\\{\rm{ = }}\sqrt {\rm{3}} \end{aligned}\)

The graph is representing the given equation.

Tangent line in point-slope

\(\begin{aligned}{c}{\rm{y - }}{{\rm{y}}_{\rm{1}}}{\rm{ = m}}\left( {{\rm{x - }}{{\rm{x}}_{\rm{1}}}} \right){\rm{:}}\\{\rm{y - rsin(}}\pi {\rm{/6) = }}\sqrt {\rm{3}} {\rm{(x - rcos(}}\pi {\rm{/6))}}\end{aligned}\)

The value of \({\rm{\theta }}\)is \(\sqrt {\rm{3}} \).