Question

find: \sin b \
ewline the figure is not drawn to scale. \
ewline answer options: \frac{36}{77}, \frac{77}{36}, \frac{36}{85}, \frac{77}{85}

Explanation:

Step1: Recall Sine Definition

In a right triangle, $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$. For angle \( B \), identify opposite and hypotenuse.
Opposite side to \( B \): \( 77 \), Hypotenuse: \( 85 \).

Step2: Apply Sine Formula

Using \(\sin B = \frac{\text{opposite to } B}{\text{hypotenuse}}\), substitute values: \(\sin B = \frac{77}{85}\)? Wait, no—wait, opposite to \( B \): Wait, the right triangle has legs 36 and 77, hypotenuse 85. Angle \( B \): adjacent is 36, opposite is 77? Wait, no, let's check the triangle. The right angle is between 36 and 77, so angle \( B \) is at the vertex with side 36. So opposite side to \( B \) is 77, hypotenuse is 85. Wait, no—wait, maybe I mixed up. Wait, in the triangle, the sides: one leg is 36 (adjacent to \( B \)), one leg is 77 (opposite to \( B \)), hypotenuse 85. So \(\sin B = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{77}{85}\)? Wait, but let's recheck. Wait, the formula is \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\). For angle \( B \), the side opposite \( B \) is 77, hypotenuse is 85. So \(\sin B = \frac{77}{85}\). Wait, but let's confirm the triangle labels. The right angle is at the corner with sides 36 and 77, so angle \( B \) is at the end of the 36 side. So opposite to \( B \) is 77, hypotenuse 85. So \(\sin B = \frac{77}{85}\). Wait, but the options: the last option is \(\frac{77}{85}\). Wait, let's check again. Wait, maybe I made a mistake. Wait, no—wait, the legs are 36 and 77, hypotenuse 85 (since \( 36^2 + 77^2 = 1296 + 5929 = 7225 = 85^2 \), so hypotenuse is 85). So for angle \( B \), opposite side is 77, hypotenuse 85. So \(\sin B = \frac{77}{85}\). Wait, but let's check the options. The fourth option is \(\frac{77}{85}\)? Wait, the options are: first \(\frac{36}{77}\), second \(\frac{77}{36}\), third \(\frac{36}{85}\), fourth \(\frac{77}{85}\). So yes, the fourth option is \(\frac{77}{85}\). Wait, but let's re-express. Wait, maybe I had the opposite wrong. Wait, angle \( B \): the side adjacent to \( B \) is 36, opposite is 77, hypotenuse 85. So \(\sin B = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{77}{85}\). So the correct option is the fourth one: \(\frac{77}{85}\).

Answer:

\(\frac{77}{85}\) (the fourth option, e.g., if options are labeled as A: \(\frac{36}{77}\), B: \(\frac{77}{36}\), C: \(\frac{36}{85}\), D: \(\frac{77}{85}\), then the answer is D. \(\frac{77}{85}\))