The exact value of cos 120 is -1/2.
Here's how to understand this:
- Unit Circle: The cosine of an angle is represented by the x-coordinate of the point where the angle intersects the unit circle.
- 120 degrees: A 120-degree angle lies in the second quadrant of the unit circle.
- Symmetry: Due to the symmetry of the unit circle, the x-coordinate of the point at 120 degrees is the negative of the x-coordinate at 60 degrees.
- Cos 60: We know that cos 60 = 1/2.
- Therefore: Cos 120 = -1/2.
Example: Imagine a triangle with angles of 30, 60, and 90 degrees. The side opposite the 60-degree angle is √3, and the hypotenuse is 2. The cosine of 60 degrees is the adjacent side (1) divided by the hypotenuse (2), which is 1/2. Since the cosine function is negative in the second quadrant, the cosine of 120 degrees is -1/2.
