Understanding and Proving cos(4π/3) -1/2 Using the Unit Circle and Right Triangle Properties

Understanding the value of trigonometric functions can be both challenging and rewarding. One common question in trigonometry is to find the value of cos(4π/3). This article will walk through a detailed proof using the unit circle and right triangle properties. We will explore how to visualize and calculate the cosine of this angle in the third quadrant.

Introduction to the Unit Circle and Trigonometric Functions

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1 centered at the origin of the coordinate plane. The x-coordinate of any point on the unit circle represents the cosine of the angle formed with the positive x-axis, while the y-coordinate represents the sine of the angle.

A Visual Approach Using the Unit Circle

Let's start by converting 4π/3 radians to degrees. We know that 4π/3 radians is equivalent to 240°. At 240°, we are in the third quadrant of the unit circle. In the third quadrant, the cosine of an angle is always negative.

In the unit circle, 240° can be visualized as:

90° beyond the 180° mark, which is the negative x-axis. The remaining 60° is the angle measured from the negative x-axis.

Therefore, cos(240°) -cos(60°). From the 30°-60°-90° triangle, we know that cos(60°) 1/2. Hence, cos(240°) -1/2, which means cos(4π/3) -1/2.

Right Triangle Approach for a Deeper Understanding

To further solidify our understanding, let's use a right triangle approach. A right triangle can be constructed with a hypotenuse of 2 units and a base of 1 unit. The angle formed at the origin can be calculated by measuring the arc along the unit circle, which we find to be π/3 or 60°. Using the definition of cosine in a right triangle, cos(π/3) 1/2.

Now, consider the angle 4π/3. This angle can be visualized as two triangles in a specific configuration:

First, draw a right triangle with an angle of π/3 (60°) and a hypotenuse of 2 units. The adjacent side to this angle is 1 unit, and the opposite side is √3 units. Since 4π/3 is in the third quadrant, we need to reflect this triangle across the y-axis to get the correct configuration. This reflection results in a base of -1 unit and a hypotenuse of 2 units, maintaining the angle of π/3 (60°). To find cos(4π/3), we use the fact that cos is an even function, cos(π - x) -cos(x). Thus, cos(4π/3) -cos(π/3) -1/2.

Proof and Verification

To verify our result, we can also use trigonometric identities. We know that cos(π - x) -cos(x). Therefore:

cos(4π/3) cos(π - π/3) -cos(π/3) -1/2

This confirms our earlier findings using the unit circle and right triangle approach.

Conclusion

In conclusion, we have proven that cos(4π/3) -1/2 using both the unit circle and right triangle methods. Each method provides a unique insight into the properties of trigonometric functions in different quadrants. Understanding these fundamental concepts is crucial for tackling more complex trigonometric problems and proofs.