Find the exact value of the trigonometric function. (If an answer is undefined, enter UNDEFINED.)
sec(25𝜋/6)

To find the exact value of \( \sec\left(\frac{25\pi}{6}\right) \), we follow these steps:

1. **Find the corresponding angle in the standard position**:

   Since \( \frac{25\pi}{6} \) is greater than \( 2\pi \), we need to find an equivalent angle between \( 0 \) and \( 2\pi \).

   \[
   \frac{25\pi}{6} - 2\pi = \frac{25\pi}{6} - \frac{12\pi}{6} = \frac{13\pi}{6}
   \]

   \(\frac{13\pi}{6}\) is still greater than \( 2\pi \). We subtract \( 2\pi \) again:

   \[
   \frac{13\pi}{6} - 2\pi = \frac{13\pi}{6} - \frac{12\pi}{6} = \frac{\pi}{6}
   \]

   Thus, \( \frac{25\pi}{6} \) is coterminal with \( \frac{\pi}{6} \).

2. **Find \( \sec\left(\frac{\pi}{6}\right) \)**:

   We know that \( \sec(\theta) = \frac{1}{\cos(\theta)} \).

   The value of \( \cos\left(\frac{\pi}{6}\right) \) is \( \frac{\sqrt{3}}{2} \).

   Therefore,

   \[
   \sec\left(\frac{\pi}{6}\right) = \frac{1}{\cos\left(\frac{\pi}{6}\right)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}
   \]

Hence, the exact value of \( \sec\left(\frac{25\pi}{6}\right) \) is:

\[
\sec\left(\frac{25\pi}{6}\right) = \frac{2\sqrt{3}}{3}
\]

Would you like further details or have any questions? 

Here are 5 related questions you can explore:

1. What is the value of \( \cos\left(\frac{25\pi}{6}\right) \)?
2. What is the value of \( \sin\left(\frac{25\pi}{6}\right) \)?
3. What is \( \sec\left(-\frac{25\pi}{6}\right) \)?
4. How do you find the period of the secant function?
5. How do you find the coterminal angle of any trigonometric function?

**Tip**: When finding the value of a trigonometric function for a large angle, first reduce it to an equivalent angle within \( 0 \) to \( 2\pi \).

Find the exact value of the trigonometric function. (If an answer is undefined, enter UNDEFINED.) sec(25π/6)

Learn how to find the exact value of sec(25π/6) by reducing it to a coterminal angle within 0 to 2π. This problem covers trigonometric concepts like the secant function and coterminal angles. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation