Question

find the exact values of the six trigonometric functions of the given angle. if any are not defined, say not defined. do not use a calculator

what is the value of \\( \cot \frac { 23 \pi } { 2 } \\) ? select the correct choice and, if necessary, fill in the answer box to complete your choice

a. \\( \cot \frac { 23 \pi } { 2 } = \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

b. the function is not defined.

what is the value of \\( \sec \frac { 23 \pi } { 2 } \\) ? select the correct choice and, if necessary, fill in the answer box to complete your choice

a. \\( \sec \frac { 23 \pi } { 2 } = \square \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

b. the function is not defined.

Explanation:

Step1: Find the coterminal angle

We know that \(2\pi\) is the period of trigonometric functions. We can find the coterminal angle of \(\frac{23\pi}{2}\) by subtracting multiples of \(2\pi\).
\(\frac{23\pi}{2}- 10\pi=\frac{23\pi - 20\pi}{2}=\frac{3\pi}{2}\)

Step2: Calculate \(\cot\frac{23\pi}{2}\)

We know that \(\cot\theta=\frac{\cos\theta}{\sin\theta}\). For \(\theta = \frac{3\pi}{2}\), \(\sin\frac{3\pi}{2}=- 1\) and \(\cos\frac{3\pi}{2}=0\). So \(\cot\frac{23\pi}{2}=\cot\frac{3\pi}{2}=\frac{\cos\frac{3\pi}{2}}{\sin\frac{3\pi}{2}}=\frac{0}{-1} = 0\)

Step3: Calculate \(\sec\frac{23\pi}{2}\)

We know that \(\sec\theta=\frac{1}{\cos\theta}\). For \(\theta=\frac{3\pi}{2}\), \(\cos\frac{3\pi}{2} = 0\). Division by zero is undefined. So \(\sec\frac{23\pi}{2}=\frac{1}{\cos\frac{3\pi}{2}}\) is not defined.

Answer:

For \(\cot\frac{23\pi}{2}\): A. \(\cot\frac{23\pi}{2}=0\)
For \(\sec\frac{23\pi}{2}\): B. The function is not defined.