Question

Use the given parameters to answer the following questions.

x = cos(3θ)
y = 5sin(θ)

(a) Find the points on the curve where the tangent is horizontal.


(b) Find the points on the curve where the tangent is vertical.



I have the answers, but for part b, 2 out of my 6 answers are wrong. Can you help me out?

Thanks in advance!

Answer 1

The equations:
\[x = \cos(3\theta)\]
\[y=5\sin(\theta)\]

Answer 2

well slope of tangent line is dy/dx

\[\frac{dy}{dx} = \frac{dy}{d \theta}*\frac{d \theta}{dx}\]
\[= 5\cos \theta * \frac{1}{-3 \sin (3 \theta)} = -\frac{5}{3} \frac{\cos \theta}{\sin (3 \theta)}\]
tangent is horizontal when slope = 0
\[\rightarrow -\frac{5 \cos \theta}{3 \sin(3 \theta)} = 0\]
this occurs when cos = 0
\[\theta = \frac{\pi}{2} + k \pi\]
plugging this in you get points (0,5) and (0,-5)

tangent is vertical when slope is infinite or undefined
this occurs when denominator approaches zero
\[\sin(3 \theta) = 0\]
\[3 \theta = k \pi\]
\[\theta =\frac{k \pi}{3}\]
plugging this in you get points (1,0) , (-1, 5sqrt3/2) , (1, 5sqrt3/2) , (-1,0), (1, -5sqrt3/2)
and (-1, -5sqrt3/2)

Answer 3

Thank You!