I have to find all the angles where the curve $r(\theta)=25+20\sin(\theta)$ has a horizontal asymptote in the interval $(0,2\pi)$, i get $\pi/2$ and $\arcsin(1)$ and $\arcsin(-1)$ but I'm either lacking some angle, or one of them is wrong?? help please?
Help with finding angles that have a horizontal tangent on a polar curve
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polar-coordinates
curves
1 Answers
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Convert to parametric cartesian with $x(\theta) = r(\theta) \, \cos\theta $ and $y(\theta) = r(\theta) \, \sin\theta $.
A horizontal tangent has ${\rm d}y = 0$ or $$ {\rm d}y = \frac{{\rm d}}{{\rm d}\theta}(r\,\sin\theta) {\rm d}\theta = (\cos\theta (25+40 \sin\theta)) {\rm d}\theta =0 $$
Solve the above for all the angles $\theta$ that make ${\rm d}y=0$