Answer:
[tex] \sf \sin(U) = \dfrac{{12}}{{13}}[/tex]
Step-by-step explanation:
To find the sine of ∠ U in triangle \triangle UTV, we use the definition of sine:
[tex] \Large\boxed{\boxed{\sf \sin(U) = \dfrac{{\textsf{Opposite}}}{{\textsf{VT}}}}}[/tex]
[tex] \sf \sin(U) = \dfrac{{\textsf{Opposite}}}{{\textsf{UV}}} [/tex]
Given:
- Adjacent side (UT)) = 20,
- Hypotenuse (UV)) = 52.
We need to find the length of the opposite side (VT). Using the Pythagorean theorem:
[tex] \sf UV^2 = UT^2 + VT^2[/tex]
[tex] \sf 52^2 = 20^2 + VT^2[/tex]
[tex] \sf 2704 = 400 + VT^2[/tex]
[tex] \sf VT^2 = 2704 - 400[/tex]
[tex] \sf VT^2 = 2304[/tex]
[tex] \sf VT = \sqrt{2304}[/tex]
[tex] \sf VT = 48[/tex]
Now, substitute the values into the formula for sine:
[tex] \sf \sin(U) = \dfrac{{48}}{{52}}[/tex]
[tex] \sf \sin(U) = \dfrac{{48 \div 4}}{{52 \div 4}}[/tex]
[tex] \sf \sin(U) = \dfrac{{12}}{{13}}[/tex]
So, the sine of ∠ U is 12/13.
