The answer is: " [tex] \frac{3}{5}[/tex] " .
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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) = [tex] \frac{3}{5}[/tex] .
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The answer is: " [tex] \frac{3}{5}[/tex] " .
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