We are asked to determine the exact value of $\cos\left(\frac{2\pi}{3}\right)$.

Step-by-Step Solution:

1. Understanding the Angle:

   • The angle $\frac{2\pi}{3}$ is in radians. To understand where this angle lies on the unit circle, we can compare it to common reference angles.
   • $\frac{2\pi}{3}$ is between $\pi/2$ and $\pi$ (i.e., it is in the second quadrant of the unit circle).

2. Reference Angle:

   • The reference angle is the smallest angle between the terminal side of $\frac{2\pi}{3}$ and the x-axis.
   • The reference angle can be found by subtracting $\frac{2\pi}{3}$ from $\pi$: $\text{Reference angle} = \pi - \frac{2\pi}{3} = \frac{3\pi}{3} - \frac{2\pi}{3} = \frac{\pi}{3}$

3. Cosine of the Reference Angle:

   • The cosine of $\frac{\pi}{3}$ is a known value from trigonometric tables or the unit circle: $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$

4. Sign of Cosine in the Second Quadrant:

   • In the second quadrant (where $\frac{2\pi}{3}$ lies), the cosine function is negative. Therefore, we take the negative of the cosine of the reference angle: $\cos\left(\frac{2\pi}{3}\right) = -\cos\left(\frac{\pi}{3}\right) = -\frac{1}{2}$

Final Answer:

$\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}$

Let me know if you want a more detailed breakdown or have any further questions! Here are some related questions for you:

1. What is the exact value of $\sin\left(\frac{2\pi}{3}\right)$?
2. How does the unit circle help in determining trigonometric values?
3. Can you find the exact value of $\cos\left(\frac{4\pi}{3}\right)$?
4. What is the significance of reference angles in trigonometry?
5. How can you derive the exact value of trigonometric functions in different quadrants?

Tip: Always remember the signs of trigonometric functions in each quadrant:

• 1st quadrant: All functions are positive.
• 2nd quadrant: Sine is positive, cosine is negative.
• 3rd quadrant: Tangent is positive, sine and cosine are negative.
• 4th quadrant: Cosine is positive, sine is negative.