Question

Make a right triangle with a right angle labeled Q and the other two acute angles labeled R and S. Afterwards, find the exact value of the ratio of cos(R) if s=15 and r=11

Answer 1

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:

\[
\cos(R) = \frac{15\sqrt{346}}{346}
\]