To find $\cos 120^\circ$ without using a calculator, we can use the following steps:

1. Recognize the quadrant: $120^\circ$ is in the second quadrant (between $90^\circ$ and $180^\circ$).

2. Reference angle: The reference angle for $120^\circ$ is $180^\circ - 120^\circ = 60^\circ$.

3. Cosine in the second quadrant: In the second quadrant, cosine is negative. So, $\cos 120^\circ = -\cos 60^\circ$.

4. Value of $\cos 60^\circ$: From basic trigonometry, we know $\cos 60^\circ = \frac{1}{2}$.

5. Final answer: $\cos 120^\circ = -\frac{1}{2}$

So, $\cos 120^\circ = -\frac{1}{2}$.

Would you like more details on any step or further questions?

Here are 5 related questions:

1. What is the reference angle for $150^\circ$?
2. How do you calculate the sine of $120^\circ$?
3. What are the cosine values in each quadrant?
4. What is the cosine of $240^\circ$?
5. How do you find trigonometric values of angles using the unit circle?

Tip: Always use the unit circle to quickly find trigonometric values for standard angles!