A30-60-90 triangle with a hypotenuse of 6 . find the length of all other sides

1.  Find the "opposite side," if the angle is 60 degrees and the hyp is 6:
     
                            opp        opp      sqrt(3)
      sin 60 deg = = =
                            hyp          6             2

      Cross-multiplying, 2(opp) = 6sqrt(3), so that opp = 3 sqrt(3) (answer)

2.  Use the Pyth. Thm. to find the "adjacent side:"

       [3sqrt(3)]^2 + x^2 = 6^2, or 9(3) + x^2 = 36, or x^2 = 9, or x = 3.

        

The lengths of the legs of this 30-60-90 triangle are 3 and 3 sqrt(3).

check:  Does 3^2 + [3sqrt(3)]^2 = 6^2?
              Does 9 + 9(3) =              6^2?   YES