Find angle y

Algebra Level 3

Two distinct angles $x , y \in [0, \dfrac{\pi}{2} ]$ are such that,

$\sin^2 x \cos x = \sin^2 y \cos y$

If $x = \dfrac{\pi}{4}$ , find $y$ .

The answer is 1.118.

1 solution

Chris Lewis
May 15, 2019

We're solving $(1-t^2)t=\frac{1}{2\sqrt2}$ , where $t=\cos y$ . Helpfully, we know that $t=\frac{1}{\sqrt2}$ is a root, so we can take out a factor of $(t-\frac{1}{\sqrt2})$ , which leaves the quadratic $t^2+\frac{1}{\sqrt2}t-\frac{1}{2}=0$ .

Solving, we find just one root in the interval $(0,1)$ , giving $\cos y=\frac{1}{4}(\sqrt{10}-\sqrt2)$ and so $y=\boxed{1.118\ldots}$ .

Find angle y

The answer is 1.118.

1 solution

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