Cos 120 Degrees Without Calculator

How to Calculate cos 120° Without a Calculator

Calculating the cosine of 120 degrees manually involves using trigonometric identities and the unit circle. Here's a step-by-step approach to find cos 120° without a calculator:

1. Identify the reference angle for 120°
2. Use the cosine of the reference angle
3. Apply the appropriate sign based on the quadrant
4. Calculate the final value

The Cosine of 120 Degrees Formula

The cosine of 120 degrees can be calculated using the following formula:

This formula comes from the fact that 120° is in the second quadrant of the unit circle, where cosine values are negative. The reference angle for 120° is 60° (180° - 120° = 60°).

Step-by-Step Calculation

1. Identify the Quadrant

   120° is in the second quadrant (90° to 180°), where cosine values are negative.

2. Find the Reference Angle

   The reference angle is calculated as 180° - 120° = 60°.

3. Use the Cosine of the Reference Angle

   We know that cos(60°) = 0.5.

4. Apply the Sign Based on the Quadrant

   Since 120° is in the second quadrant, we take the negative of the cosine of the reference angle.

5. Calculate the Final Value

   cos(120°) = -cos(60°) = -0.5

Worked Example

Let's calculate cos 120° step by step:

1. Determine the quadrant: 120° is in the second quadrant (90° < 120° < 180°).
2. Find the reference angle: 180° - 120° = 60°.
3. Recall that cos(60°) = 0.5.
4. Apply the sign: Since cosine is negative in the second quadrant, cos(120°) = -0.5.

The final result is cos(120°) = -0.5.

Interpreting the Result

The cosine of 120 degrees is -0.5. This means:

• The x-coordinate on the unit circle is -0.5
• The angle forms a 60° angle with the negative x-axis
• In a right triangle with angle 120°, the adjacent side is half the length of the hypotenuse