value of cos 120

Value of cos 120°: An Academic Explanation

In trigonometry, the cosine (denoted as cos) is a fundamental trigonometric function that relates the angle of a right-angled triangle to the ratio of the adjacent side to the hypotenuse. It is one of the key functions used to study the properties of angles, especially in the unit circle, which is a circle with a radius of 1 centered at the origin of a coordinate plane.

In this article, we will explore the value of cos 120°, explain how to find it, and its significance in both right-angled triangles and the unit circle.

Understanding cos 120° Using the Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. It helps us determine the values of trigonometric functions like sine, cosine, and tangent for any angle.

To find cos 120°, let’s consider the following steps:

1. Locate 120° on the Unit Circle: Angles are measured counterclockwise from the positive x-axis. The angle 120° lies in the second quadrant of the unit circle (between 90° and 180°).
2. Determine the Reference Angle: The reference angle is the smallest angle formed by the terminal side of the given angle and the x-axis. For 120°, the reference angle is:
   $180° - 120° = 60°$
3. Cosine Value in the Second Quadrant: In the second quadrant, the cosine value is negative because the x-coordinate of points on the unit circle in this quadrant is negative. The cosine of an angle is the x-coordinate of the point on the unit circle corresponding to that angle.
4. Value of cos 60°: From trigonometric tables or the unit circle, we know that: $\cos 60° = \frac{1}{2}$
5. Cosine of 120°: Since 120° is in the second quadrant, and the reference angle is 60°, the value of cos 120° is the negative of cos 60°. Hence: $\cos 120° = -\frac{1}{2}$

The Final Answer

Thus, the value of cos 120° is: $\cos 120° = -\frac{1}{2}$

This means that the cosine of 120° is negative, and its value is -0.5.

Conclusion

In summary, using the unit circle, we can determine that the value of cos 120° is -1/2 or -0.5. This result comes from the fact that cosine represents the x-coordinate on the unit circle, and in the second quadrant, the x-coordinate is negative. Understanding this concept is essential in trigonometry as it helps in solving various mathematical and real-world problems involving angles and trigonometric functions.