If $\tan\theta=\frac{12}{5}$, find the values of the five other trigonometric functions. Rationalize the denominators if necessary.

• A. $\sin\theta=\frac{12}{13},\cos\theta=\frac{5}{13},\cot\theta=\frac{5}{12},\sec\theta=\frac{13}{5},\csc\theta=\frac{13}{12}$
• B. $\sin\theta=\frac{5}{13},\cos\theta=\frac{12}{13},\cot\theta=\frac{5}{12},\sec\theta=\frac{13}{12},\csc\theta=\frac{13}{5}$
• C. $\sin\theta=\frac{12}{13},\cos\theta=\frac{5}{13},\cot\theta=-\frac{5}{12},\sec\theta=-\frac{13}{5},\csc\theta=-\frac{13}{12}$
• D. $\sin\theta=\frac{5}{13},\cos\theta=\frac{12}{13},\cot\theta=-\frac{5}{12},\sec\theta=-\frac{13}{12},\csc\theta=-\frac{13}{5}$

Answer: (A)

Verified step by step guidance

1. Start by recalling the definition of the tangent function: \( \tan\theta = \frac{\text{opposite}}{\text{adjacent}} \). Given \( \tan\theta = \frac{12}{5} \), we can consider a right triangle where the opposite side is 12 and the adjacent side is 5.
2. Use the Pythagorean theorem to find the hypotenuse of the triangle. The theorem states \( a^2 + b^2 = c^2 \), where \( a \) and \( b \) are the legs of the triangle, and \( c \) is the hypotenuse. Substitute the known values: \( 12^2 + 5^2 = c^2 \).
3. Solve for \( c \) to find the hypotenuse. This will give you the length of the hypotenuse, which is necessary to find the sine and cosine of \( \theta \).
4. Calculate \( \sin\theta \) and \( \cos\theta \) using the definitions: \( \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \) and \( \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \). Substitute the values from the triangle.
5. Determine the remaining trigonometric functions: \( \cot\theta = \frac{1}{\tan\theta} \), \( \sec\theta = \frac{1}{\cos\theta} \), and \( \csc\theta = \frac{1}{\sin\theta} \). Rationalize the denominators if necessary.