Given a right triangle where angle $\theta$ is one of the non-right angles, the side opposite $\theta$ has length $3$, the side adjacent to $\theta$ has length $4$, and the hypotenuse has length $5$, what is the value of $tan\left(\theta \right)$?

• A. $\frac{3}{4}$
• B. $\frac{4}{3}$
• C. $\frac{3}{5}$
• D. $\frac{4}{5}$

Answer: (A)

Verified step by step guidance

1. Identify the sides of the right triangle relative to the angle \( \theta \): the side opposite \( \theta \) is 3, the side adjacent to \( \theta \) is 4, and the hypotenuse is 5.
2. Recall the definition of the tangent function in a right triangle: \( \tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} \).
3. Substitute the given side lengths into the tangent formula: \( \tan(\theta) = \frac{3}{4} \).
4. Understand that the tangent value represents the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
5. Conclude that the value of \( \tan(\theta) \) is the fraction \( \frac{3}{4} \), which corresponds to the ratio of the opposite side to the adjacent side.