Determining the Exact Value of tan 120°: A Comprehensive Guide

Understanding Trigonometric Functions: Trigonometric functions, such as tangent (tan), are fundamental in mathematics and have numerous applications in various fields including engineering, physics, and architecture. The value of a trigonometric function can often be determined based on the angle and the quadrant in which it lies.

tan 120° Simplified: The tangent of an angle can be determined by expressing it in terms of the sine and cosine of the angle, as tan X sin X / cos X. Specifically, for the case of 120°, its value can be derived using the following identity and properties of the unit circle.

Deriving tan 120° Using Fundamental Identities

Method 1: Direct Trigonometric Identities

To find the exact value of tan 120°, we can use the fundamental identity:

tan X sin X / cos X sin 120° / cos 120° (sin 180° - 60°) / (cos 180° - 60°) sin 60° / -cos 60° (√3/2) / (-1/2) -√3.

Method 2: Secant and Sine Identity

Another approach is to utilize the identity sec X 1 / cos X, along with the sine of the angle:

tan 120° sec 120° * sin 120°. Given that sec 120° -2 and sin 120° √3/2, we can calculate:

tan 120° -2 * (√3/2) -√3.

Referring to the Unit Circle

Considering 120° is 60° above the X-axis in the 2nd quadrant, and knowing that sin 120° sin 60° and cos 120° -cos 60°, we can directly state:

sin 120° √3/2 and cos 120° -1/2, so tan 120° sin 120° / cos 120° -√3.

Double Angle Identity

Another method involves using the double angle identity for tangent, tan 2x 2tan x / (1 - tan^2 x). For 120°, which is 2 * 60°, we have:

tan 120° 2 * tan 60° / (1 - tan^2 60°) 2 * √3 / (1 - 3) 2 * √3 / (-2) -√3.

Angle Transformation

By transforming 120° to 180° - 60°, we can use the identity tan(180° - x) -tan x, which gives:

tan 120° tan(180° - 60°) -tan 60° -√3.

Summary and Final Answer

Considering the position of 120° in the second quadrant and the signs of sine and cosine, we can conclude:

120° is in the 2nd quadrant, with a reference angle of 60°. Since tangent is negative in the 2nd quadrant, we have:

tan 120° -tan 60° -√3.

Conclusion

This comprehensive guide has detailed the various methods to find the exact value of tan 120°. By utilizing trigonometric identities, properties of the unit circle, and angle transformations, it is clear that the value of tan 120° is -√3. This understanding is essential for solving more complex trigonometric equations and problems in mathematics and related fields.

Keywords: trigonometric functions, tangent, values of trigonometric functions