To create a right triangle with a right angle at \( Q \) and acute angles \( R \) and \( S \), we can visualize it as follows:
S
/|
/ |
/ |
15 / | r (opposite to angle S)
/ |
/ |
/______ |
Q R
In this triangle:
- The side opposite angle \( R \) (which is adjacent to \( S \)) is \( r = 11 \).
- The side opposite angle \( S \) (which is adjacent to \( R \)) is \( s = 15 \).
- We can use the Pythagorean theorem to find the hypotenuse \( h \):
\[ h = \sqrt{s^2 + r^2} = \sqrt{15^2 + 11^2} = \sqrt{225 + 121} = \sqrt{346} \]
Now, to find \( \cos(R) \):
The cosine of angle \( R \) in a right triangle is defined as the ratio of the length of the adjacent side (which is \( s = 15 \)) to the hypotenuse \( h \):
\[ \cos(R) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{s}{h} = \frac{15}{\sqrt{346}} \]
To express this in a more standard form (i.e., without the square root in the denominator), we can multiply the numerator and denominator by \( \sqrt{346} \):
\[ \cos(R) = \frac{15}{\sqrt{346}} \cdot \frac{\sqrt{346}}{\sqrt{346}} = \frac{15\sqrt{346}}{346} \]
Thus, the exact value of the ratio of \( \cos(R) \) is:
\[ \cos(R) = \frac{15\sqrt{346}}{346} \]