If the legs of a right triangle are ' x ' and ' x³ ' , the the hypotenuse is
√[ (x)² + (x³)² ]
= √ (x² + x⁶)
= √ x²(1 + x⁴) .
The cosine of one acute angle is
x / x²(1 + x⁴) = 1 / x(1 + x⁴) .
The cosine of the other acute angle is
x³ / x²(1 + x⁴) = x / (1 + x⁴) .
Te value of the cosine function is 1 when the angle=0, and it drops steadily to a value of 0 zero when the angle is 90°. So the smaller angle has the larger cosine.
There's a little tiny problem here . . . . . we don't really know which cosine is larger, because we don't know whether 'x' is less than 1 or more than 1.
It's past my bedtime, and I'm not able to unravel all the possibilities. Let's just assume that 'x' is greater than 1. In that case, the larger cosine is
<span>1 / x(1 + x⁴)
(I think), so that's the cosine of the smaller acute angle.</span>
