Let's analyze the given triangle [tex]\( \triangle RST \)[/tex] using the law of cosines for different scenarios provided in the options.
Using the information from the question:
[tex]\[
r = 5, \quad s = 7, \quad t = 3
\][/tex]
We start by applying the law of cosines in the form:
[tex]\[
r^2 = s^2 + t^2 - 2st \cos(S)
\][/tex]
Plugging in the values:
[tex]\[
5^2 = 7^2 + 3^2 - 2(7)(3) \cos(S)
\][/tex]
First, calculate each side:
[tex]\[
5^2 = 25
\][/tex]
[tex]\[
7^2 = 49
\][/tex]
[tex]\[
3^2 = 9
\][/tex]
[tex]\[
2 \cdot 7 \cdot 3 = 42
\][/tex]
Now substitute these values back into our equation:
[tex]\[
25 = 49 + 9 - 42 \cos(S)
\][/tex]
Simplify the right side:
[tex]\[
25 = 58 - 42 \cos(S)
\][/tex]
To isolate [tex]\(\cos(S)\)[/tex], subtract 58 from both sides:
[tex]\[
25 - 58 = -42 \cos(S)
\][/tex]
This simplifies to:
[tex]\[
-33 = -42 \cos(S)
\][/tex]
Now, divide both sides by -42 to solve for [tex]\(\cos(S)\)[/tex]:
[tex]\[
\cos(S) = \frac{-33}{-42}
\][/tex]
Simplify the fraction:
[tex]\[
\cos(S) = \frac{33}{42} = \frac{11}{14} \approx 0.7857
\][/tex]
So, the cosine of angle [tex]\(S\)[/tex] is approximately 0.7857. Thus, the value cos [tex]\(S\)[/tex] we have here corroborates with the given numerical result.
Based on the triangle sides provided (5, 7, and 3), we used all sides correctly in our calculations according to the law of cosines. This confirms that the triangle sides [tex]\(r = 5, s = 7, t = 3 \)[/tex] are correct for the given problem and satisfy the law of cosines.
Therefore, the correct statement about [tex]\( \triangle RST\)[/tex] is:
- [tex]\( r = 5\)[/tex]
- [tex]\( s = 7\)[/tex]
- [tex]\( t = 3 \)[/tex]
