Given a right triangle where the side opposite angle $\theta$ has length $3$, the adjacent side has length $4$, and the hypotenuse has length $5$, what is the equation for the trigonometric function $f\left(x\right)$ that represents $sin$ of angle $\theta$ in terms of these side lengths?

• A. $f\left(x\right)=\frac{4}{5}$
• B. $f\left(x\right)=\frac{3}{5}$
• C. $f\left(x\right)=\frac{3}{4}$
• D. $f\left(x\right)=\frac{5}{3}$

Answer: (B)

Verified step by step guidance

1. Identify the sides of the right triangle relative to angle \( \theta \): the side opposite \( \theta \) is 3, the adjacent side is 4, and the hypotenuse is 5.
2. Recall the definition of the sine function in a right triangle: \( \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \).
3. Substitute the given side lengths into the sine formula: \( \sin(\theta) = \frac{3}{5} \).
4. Express the trigonometric function \( f(x) \) that represents \( \sin(\theta) \) as \( f(x) = \frac{3}{5} \).
5. Verify that this expression matches the correct ratio of the opposite side to the hypotenuse, confirming the function \( f(x) = \sin(\theta) = \frac{3}{5} \).