Question

The legs of a right triangle are lengths x and x3. The cosine of the smallest angle of the triangle is what

Answer 1

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

                                               1 / x(1 + x⁴)

(I think), so that's the cosine of the smaller acute angle.