Maths Olympiad Prep

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Algebra Difficulty 4.8 AIME Find the answer

11. Let x,yx, y be acute angles, and x+y=60∘,x+y=60^{\circ}, the maximum value of sin⁡x+sin⁡y\sin x+\sin y is

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

Solution

11. 1.

Since sin⁡x+sin⁡y=2sin⁡x+y2cos⁡x−y2=cos⁡x−y2=cos⁡(x−30∘),x∈(0∘,60∘)\sin x+\sin y=2 \sin \frac{x+y}{2} \cos \frac{x-y}{2}=\cos \frac{x-y}{2}=\cos \left(x-30^{\circ}\right), x \in\left(0^{\circ}, 60^{\circ}\right), the maximum value of sin⁡x+sin⁡y\sin x+\sin y is 1.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.