# Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

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Out Of Place Trig. Topic: Geometry/Trigonometry. Level: AMC/AIME. October 29th, 2006

Problem: (2006-2007 Warm-Up 4) Two sides of a triangle are and and the angle between them is . If , what is the maximum area of the triangle?

Solution: Well, given two sides and the angle between them, there is only one formula for the area of a triangle that comes to mind. . So plugging this in, we get the area to be

.

By the double-angle identity, this is equal to

for . But the maximum of this is clearly and it is obtained when . QED.

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Comment: This problem shouldn’t have been too hard; finding that specific formula for the area of the triangle could have been derived by drawing an altitude. And the double-angle identity should always be known. Lastly, maximizing the sine function is a given.

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Practice Problem: (2006 Skyview) If and , what is ?

### One Response to “Out Of Place Trig. Topic: Geometry/Trigonometry. Level: AMC/AIME.”

1. mysmartmouth Says:

Practice Problem:

Use De Moivre’s: cis(3x) = (cis x)^3

Multiply out and equate imaginary parts:

sin 3x = 3 cos^2 x sin x – sin^3 x

Thus sin 3x = 117/125 from the given values.