Given a right triangle where angle $a$ has an opposite side of length $3$, an adjacent side of length $4$, and a hypotenuse of length $5$, use the triangle to evaluate each function: $\sin \left(a\right)$, $\tan \left(a\right)$, and $\sec \left(a\right)$.

• A. $\sin \left(a\right)=\frac{5}{3},\tan \left(a\right)=\frac{3}{5},\sec \left(a\right)=\frac{4}{5}$
• B. $\sin \left(a\right)=\frac{3}{4},\tan \left(a\right)=\frac{5}{3},\sec \left(a\right)=\frac{4}{3}$
• C. $\sin \left(a\right)=\frac{4}{5},\tan \left(a\right)=\frac{4}{3},\sec \left(a\right)=\frac{5}{3}$
• D. $\sin \left(a\right)=\frac{3}{5},\tan \left(a\right)=\frac{3}{4},\sec \left(a\right)=\frac{5}{4}$

Answer: (D)

Verified step by step guidance

1. Identify the sides of the right triangle relative to angle \(a\): the opposite side is 3, the adjacent side is 4, and the hypotenuse is 5.
2. Recall the definitions of the trigonometric functions in a right triangle: \(\sin(a) = \frac{\text{opposite}}{\text{hypotenuse}}\), \(\tan(a) = \frac{\text{opposite}}{\text{adjacent}}\), and \(\sec(a) = \frac{\text{hypotenuse}}{\text{adjacent}}\).
3. Calculate \(\sin(a)\) by dividing the length of the opposite side by the hypotenuse: \(\sin(a) = \frac{3}{5}\).
4. Calculate \(\tan(a)\) by dividing the length of the opposite side by the adjacent side: \(\tan(a) = \frac{3}{4}\).
5. Calculate \(\sec(a)\) by dividing the length of the hypotenuse by the adjacent side: \(\sec(a) = \frac{5}{4}\).