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