Question

What is the trigonometric ratio for sin Z?

Answer 1

The answer is: "
`(3)/(5)` " .
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Step-by-step 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) =
`(3)/(5)` .
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The answer is: "
`(3)/(5)` " .
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