Question
In \( \triangle Q R S \), the measure of \( \angle S=90^{\circ}, R Q=61, S R=11 \), and \( Q S=60 \). What ratio represents the sine of \( \angle R \) ?
Ask by Campos Klein. in the United States
Jan 27,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The sine of \( \angle R \) is \( \frac{60}{61} \).
Solution
In right triangle \( \triangle QRS \) with \( \angle S = 90^\circ \), we have the following sides:
- **Hypotenuse** (\( QR \)) = 61
- **Leg opposite \( \angle Q \)** (\( RS \)) = 11
- **Leg opposite \( \angle R \)** (\( QS \)) = 60
To find \( \sin(\angle R) \), use the definition of sine in a right triangle:
\[
\sin(\angle R) = \frac{\text{Opposite side to } \angle R}{\text{Hypotenuse}} = \frac{QS}{QR} = \frac{60}{61}
\]
**Therefore, the ratio representing the sine of \( \angle R \) is \( \frac{60}{61} \).**
\(\boxed{\dfrac{60}{61}}\)
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Simplify this solution Mind Expander
To find the sine of \( \angle R \) in triangle \( \triangle QRS \) with the given sides, we can use the definition of sine in a right triangle. We know that: \[ \sin R = \frac{\text{opposite side to } \angle R}{\text{hypotenuse}} \] In our triangle, the side opposite to \( \angle R \) is \( S Q \) (which is 60), and the hypotenuse is \( R Q \) (which is 61). Therefore, we can write: \[ \sin R = \frac{Q S}{R Q} = \frac{60}{61} \] Thus, the ratio that represents the sine of \( \angle R \) is \( \frac{60}{61} \).
