Prove that the sum of the legs of a right triangle never exceeds √2 times the hypotenuse.
The ratio of a+b to c will not change if we resize the triangle so c=1. Now place the triangle so we're actually dealing with the unit circle. For angle a and we need to prove sin(a)+cos(a) <= sqrt2
From the sum to product rule sin(a)+cos(a)
= sin(a)+sin(pi/2-a)
=2sin(pi/4)cos(a-pi/4)
=sqrt(2)sin(a+pi/4)
which clearly never exceeds sqrt(2).
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Posted by Jer
on 2024-11-22 15:10:44 |
Trig solution