Answer:
12/13
Step-by-step explanation:
To find [tex]\sf \sin(\angle W) [/tex], we proceed as follows:
Calculate the Hypotenuse [tex]\sf VW [/tex]
In a right-angled triangle, the Pythagorean theorem states that:
[tex]\sf \textsf{Hypotenuse}^2 = (\textsf{Opposite side})^2 + (\textsf{Adjacent side})^2 [/tex]
Here, the opposite side [tex]\sf VX [/tex] is 48, and the adjacent side [tex]\sf WX [/tex] is 20. Therefore:
[tex]\sf VW^2 = VX^2 + WX^2 [/tex]
[tex]\sf VW^2 = 48^2 + 20^2 [/tex]
[tex]\sf VW^2 = 2304 + 400 [/tex]
[tex]\sf VW^2 = 2704 [/tex]
Taking the square root of both sides:
[tex]\sf VW = \sqrt{2704} [/tex]
[tex]\sf VW = 52 [/tex]
So, the hypotenuse [tex]\sf VW [/tex] is [tex]\sf 52 [/tex].
Now,
Calculate [tex]\sf \sin(\angle W) [/tex]
The sine of an angle in a right-angled triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. Thus:
[tex]\sf \sin(\angle W) = \dfrac{\textsf{Opposite}}{\textsf{Hypotenuse}} [/tex]
For angle [tex]\sf W [/tex]:
[tex]\sf \sin(\angle W) = \dfrac{VX}{VW} [/tex]
[tex]\sf \sin(\angle W) = \dfrac{48}{52} [/tex]
Simplify the fraction:
[tex]\sf \sin(\angle W) = \dfrac{48 \div 4}{52 \div 4} [/tex]
[tex]\sf \sin(\angle W) = \dfrac{12}{13} [/tex]
Therefore, the sin of [tex]\sf \angle W [/tex] is 12/13.
