Find the maximum y

Calculus Level pending

Find the maximum y coordinate of a point on the graph of $r = \sin 2 \theta.$


The answer is 0.7698.

Aly Ahmed by Aly Ahmed , 34, Egypt

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1 solution

We have y = r sin ⁡ θ = sin ⁡ 2 θ sin ⁡ θ = 2 sin ⁡ 2 θ cos ⁡ θ y=r\sin \theta=\sin 2\theta\sin \theta=2\sin^2 \theta\cos \theta .

This will be maximum when it's first derivative with respect to θ \theta is zero, and it's second derivative is negative. This implies cos ⁡ θ = 1 3 , sin ⁡ θ = 2 3 ⟹ y m a x = 2 × 2 3 × 1 3 = 4 3 3 ≈ 0.7698 \cos \theta=\dfrac{1}{\sqrt 3}, \sin \theta=\sqrt {\dfrac{2}{3}}\implies y_{max}=2\times \dfrac{2}{3}\times \dfrac{1}{\sqrt 3}=\dfrac{4}{3\sqrt 3}\approx \boxed {0.7698} .

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what it meant by the statement: $r = \sin 2 \theta.$ I am unable to interpret the question.

Pradeep Tripathi - 1 year, 1 month ago

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It's r = sin ⁡ 2 θ r=\sin 2\theta

A Former Brilliant Member - 1 year, 1 month ago

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