Question

find all points on r=2sinθ where the tangent line is parallel to the ray θ=π/4

Answer 1

(2sin pi/8,pi/8),(2 sin 5pi/8,5pi/8) are the answers if it helps!

Answer 2

\[

\frac{dy}{dx}=\frac{\cos (\theta ) r(\theta )+\sin (\theta ) \frac{\partial r(\theta
)}{\partial \theta }}{\cos (\theta ) \frac{\partial r(\theta
)}{\partial \theta }-\sin (\theta ) r(\theta )}=\frac{4 \sin (\theta )
\cos (\theta )}{2 \cos ^2(\theta )-2 \sin ^2(\theta )}
\]
Set
\[
\frac{4 \sin (\theta )
\cos (\theta )}{2 \cos ^2(\theta )-2 \sin ^2(\theta )}
\]=1

and solve for \(\theta\)
you find
\[
\theta =\frac \pi 8\\
\theta =\frac {5\pi}8
\]

Answer 3

See http://tutorial.math.lamar.edu/Classes/CalcII/PolarTangents.aspx

Answer 4

\[
\frac{4 \sin (\theta ) \cos (\theta )}{2 \cos ^2(\theta )-2 \sin ^2(\theta
)}=1\\
4 \sin (\theta ) \cos (\theta )=2 \cos ^2(\theta )-2 \sin ^2(\theta )=2\cos(2 \theta)\\
2 \sin(2\theta)= 2\cos(2 \theta)\\
\sin(2\theta)= \cos(2 \theta)\\
\]
It is easy from now on

Answer 5

@ganeshie8

Answer 6

To finish your problem above, us

\[
\sin(x)=\sin(y)\\
x=\frac \pi 4\\
x= \frac {5\pi}4\\
0\le x \le 2\pi

\]

Answer 7

Ahh nice :) I almost forgot about that dy/dx formula lol..
I took the hard way by converting it to cartesian and got \(\large y = 1 \pm\frac{1}{\sqrt{2}}\) ... again i need to switch it back to polar which is bit of pain :/
thanks @eliassaab for showing the work :)

Answer 8

Thank you guys :)

Answer 9

Why do you set the tangent line equal to one?

Answer 10

do u mean, why the "slope of tangent line" was set equal to 1 ?

Answer 11

yes

Answer 12

read the question once again :
... the tangent line is parallel to the ray θ=π/4

Answer 13

whats the slope of ray : θ=π/4 ?

Answer 14

oh alrights thanks!

Answer 15

:)