Given a right triangle where one of the acute angles is $\theta$, and the side opposite $\theta$ has length $3$, the side adjacent to $\theta$ has length $4$, and the hypotenuse has length $5$, what are the exact values of the six trigonometric ratios for angle $\theta$?

• A. $\sin \theta =\frac{3}{5},\cos \theta =\frac{4}{5},\tan \theta =\frac{3}{4},\csc \theta =\frac{5}{3},\sec \theta =\frac{5}{4},\cot \theta =\frac{4}{3}$
• B. $\sin \theta =\frac{4}{5},\cos \theta =\frac{3}{5},\tan \theta =\frac{4}{3},\csc \theta =\frac{5}{4},\sec \theta =\frac{5}{3},\cot \theta =\frac{3}{4}$
• C. $\sin \theta =\frac{3}{4},\cos \theta =\frac{4}{5},\tan \theta =\frac{3}{5},\csc \theta =\frac{4}{3},\sec \theta =\frac{5}{4},\cot \theta =\frac{5}{3}$
• D. $\sin \theta =\frac{4}{5},\cos \theta =\frac{3}{4},\tan \theta =\frac{5}{3},\csc \theta =\frac{4}{5},\sec \theta =\frac{3}{5},\cot \theta =\frac{5}{4}$

Answer: (A)

Verified step by step guidance

1. Identify the sides of the right triangle relative to angle \( \theta \): the side opposite \( \theta \) is 3, the side adjacent to \( \theta \) is 4, and the hypotenuse is 5.
2. Recall the definitions of the six trigonometric ratios for angle \( \theta \): - \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \) - \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \) - \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \) - \( \csc \theta = \frac{1}{\sin \theta} = \frac{\text{hypotenuse}}{\text{opposite}} \) - \( \sec \theta = \frac{1}{\cos \theta} = \frac{\text{hypotenuse}}{\text{adjacent}} \) - \( \cot \theta = \frac{1}{\tan \theta} = \frac{\text{adjacent}}{\text{opposite}} \)
3. Substitute the given side lengths into each ratio: - \( \sin \theta = \frac{3}{5} \) - \( \cos \theta = \frac{4}{5} \) - \( \tan \theta = \frac{3}{4} \) - \( \csc \theta = \frac{5}{3} \) - \( \sec \theta = \frac{5}{4} \) - \( \cot \theta = \frac{4}{3} \)
4. Verify that the values are consistent with the Pythagorean theorem and the properties of trigonometric functions for acute angles in a right triangle.
5. Summarize the exact values of the six trigonometric ratios for angle \( \theta \) using the fractions derived from the side lengths.