HMMT February

Algebra Difficulty 5.6 AIME, harder Find the answer

Alex picks his favorite point $(x, y)$ in the first quadrant on the unit circle $x^{2}+y^{2}=1$ , such that a ray from the origin through $(x, y)$ is $\theta$ radians counterclockwise from the positive $x$ -axis. He then computes $\cos ^{-1}\left(\frac{4 x+3 y}{5}\right)$ and is surprised to get $\theta$ . What is $\tan (\theta)$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

$x=\cos (\theta), y=\sin (\theta)$ . By the trig identity you never thought you'd need, $\frac{4 x+3 y}{5}=\cos (\theta-\phi)$ , where $\phi$ has sine $3 / 5$ and cosine $4 / 5$ . Now $\theta-\phi=\theta$ is impossible, since $\phi \neq 0$ , so we must have $\theta-\phi=-\theta$ , hence $\theta=\phi / 2$ . Now use the trusty half-angle identities to get $\tan (\theta)=\frac{1}{3}$ .

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