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 understanding that \( \tan\theta = \frac{12}{5} \) represents the ratio of the opposite side to the adjacent side in a right triangle. This means the opposite side is 12 units and the adjacent side is 5 units.
2. Use the Pythagorean theorem to find the hypotenuse. The formula is \( c = \sqrt{a^2 + b^2} \), where \( a = 12 \) and \( b = 5 \). Calculate \( c = \sqrt{12^2 + 5^2} \).
3. Now that you have the hypotenuse, you can find \( \sin\theta \) and \( \cos\theta \). \( \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \) and \( \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \). Substitute the values to find \( \sin\theta \) and \( \cos\theta \).
4. To find \( \cot\theta \), use the reciprocal of \( \tan\theta \). Since \( \tan\theta = \frac{12}{5} \), \( \cot\theta = \frac{5}{12} \).
5. Finally, find \( \sec\theta \) and \( \csc\theta \) using the reciprocals of \( \cos\theta \) and \( \sin\theta \) respectively. \( \sec\theta = \frac{1}{\cos\theta} \) and \( \csc\theta = \frac{1}{\sin\theta} \). Rationalize the denominators if necessary.