$$ Find:{Cos}\mathbf{120}\degree withoutusingcalculator $$

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!

Find: $ \cos 120^\circ $ without using a calculator

Learn how to find the cosine of 120 degrees without a calculator. This solution explains key concepts such as reference angles, the cosine function, and quadrant rules. Perfect for high school students studying trigonometry, suitable for Grades 9-12.

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