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