What is the trigonometric ratio for sin z?

The answer is:  "  " .
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Explanation:
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Using:  "SOH CAH TOA" ;

Note that "SOH" applies; since we are dealing with the "sin" ;

         →   "sin = opp/ hyp" ;

that is:  sin = opposite / hypotenuse.
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Thus, "sin Z = opposite side / hypotenuse" .
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From the figure provided, we see that the hypotenuse is:  "40".

However, the "opposite" side (with respect to "angle Z") ; which is side "XY" ; is not provided.

So, we can solve for the "opposite side", which is side "XY" ; 

using the "Pythogorean theorem" ; which is the equation/formula for the sides of a right triangle;
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    which is: 
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 →  a² + b² = c² ;

{Note: The right angle in the particular triangle of concern is "angle Y" ;

and side  "a" is "XY" ; for which we wish to solve; 

                "b" = 32 (as shown in figure);

                "c" = hypotenuse" = 40 (as shown in figure) .}.
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    a² + b² = c²  ;

 → a² = c² - b² ;

     a² = 40² − 32² ;

     a² = (40*40) − (32*32) ;

     a² = (1600) − (1024) ;

     a² = 576 ;

Take the positive square root of EACH SIDE of the equation; to isolate "a" on one side of the equation; & to solve for "a" ; 

   +√(a²) = +√576 ;

        a = 24 ;
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SO;  "XY" = 24 ;

So, using "SOH" ; 

sin Z = opposite / hypotenuse ; 
The "opposite" is "XY" = 24 .  The hypotenuse = 40; 

So;  sin Z = 24 / 40 ;

which can be simplified as follows:

  24/40 = (24÷8) / (40÷8) = .
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The answer is:  "  " .
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