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What is the measure of the smallest of two acute angle?

What is the measure of the largest of two acute angle?

The real answer is we don't know; really the best we can do is an approximation with a calculator.  They're asking for that approximation, but don't think for a second that's the actual exact measure of the angles here.  It just goes to show we don't need angles; we have these exact lengths -- why mush it up with an inexact angle?

But we do what we're told.  In the a=8, b=15, c=17 right triangle we have two acute angles, of course complementary, adding to 90 degrees. The usual right triangle identities tell us

sin A = cos B = a/c, cos A = sin B = b/c, tan A = cot B = a/b, tan B = cot A = b/a.

so

A = arcsin(8/17) = arccos(15/17) = arctan(8/15)

B = arcsin(15/17) = arccos(8/17) = arctan(15/8)

We've written about all the exact things we can write.  

A = arctan(8/15) ≈ 28.07°  (smallest)

B = arctan(15/8) ≈ 61.93°   (largest)