Track / Stage 5 / 137 of 400 #737 of 1964

Problem 737

AIME late

Algebra Difficulty 5.2 Find the answer

[4] Let $a \star b=\sin a \cos b$ for all real numbers $a$ and $b$ . If $x$ and $y$ are real numbers such that $x \star y-y \star x=1$ , what is the maximum value of $x \star y+y \star x$ ?

Your answer

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

Official solution

Answer: $\square$
We have $x \star y+y \star x=\sin x \cos y+\cos x \sin y=\sin (x+y) \leq 1$ . Equality is achieved when $x=\frac{\pi}{2}$ and $y=0$ . Indeed, for these values of $x$ and $y$ , we have $x \star y-y \star x=\sin x \cos y-\cos x \sin y=\sin (x-y)=$ $\sin \frac{\pi}{2}=1$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.