Find the exact value of the trigonometric function. (If an answer is undefined, enter UNDEFINED.)

cos(4𝜋/3)

To find the exact value of \( \cos\left(\frac{4\pi}{3}\right) \), we proceed with the following steps:

1. **Locate the angle on the unit circle**: 
   The angle \( \frac{4\pi}{3} \) is in radians. Converting to degrees, \( \frac{4\pi}{3} \times \frac{180^\circ}{\pi} = 240^\circ \).
   
2. **Identify the quadrant**: 
   An angle of \( 240^\circ \) lies in the third quadrant of the unit circle, where cosine values are negative.

3. **Find the reference angle**:
   The reference angle for \( 240^\circ \) is \( 240^\circ - 180^\circ = 60^\circ \) (or \( \frac{\pi}{3} \) radians).

4. **Evaluate cosine**:
   The cosine of the reference angle \( 60^\circ \) is \( \cos(60^\circ) = \frac{1}{2} \). Since cosine is negative in the third quadrant, we have:
   \[
   \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2}.
   \]

Thus, the exact value is:
\[
\boxed{-\frac{1}{2}}.
\]

Would you like further details on the process?

Here are 5 related questions:
1. What is the sine value of \( \frac{4\pi}{3} \)?
2. How do you determine reference angles for trigonometric functions?
3. What is the value of \( \tan\left(\frac{4\pi}{3}\right) \)?
4. How can you use symmetry to find values of cosine for any angle?
5. What is the general rule for determining the sign of trigonometric functions in different quadrants?

**Tip**: When working with trigonometric functions, always consider the quadrant to determine whether the value is positive or negative.

Find the exact value of the trigonometric function. (If an answer is undefined, enter UNDEFINED.) cos(4π/3)

Discover the step-by-step process to find the exact value of cos(4π/3) using the unit circle. Learn how to locate the angle, determine the reference angle, and apply quadrant rules to evaluate cosine. Perfect for high school students studying trigonometry in Grades 10-12.

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