Equivalent Expression To Cos 120 Degrees

#Introduction

In the realm of trigonometry, understanding trigonometric functions and their properties is paramount. One fundamental concept is identifying equivalent expressions for trigonometric values, particularly within different quadrants of the unit circle. The question at hand, "Which expression is equivalent to cos 120°?", delves into this very concept. This article will thoroughly explore the solution by examining the properties of the cosine function, reference angles, and the symmetry inherent in the unit circle. This exploration aims to provide a clear, step-by-step understanding, ensuring clarity on why certain options are equivalent while others are not. We will dissect each potential answer, providing a robust mathematical explanation that leaves no room for ambiguity. Grasping these principles is not just about answering this specific question; it's about building a solid foundation in trigonometry, which is crucial for success in higher mathematics and related fields. So, let's embark on this trigonometric journey and unravel the solution together, solidifying your understanding of cosine functions and their equivalencies.

Understanding the Cosine Function and the Unit Circle

To effectively determine which expression is equivalent to cos 120°, a solid grasp of the cosine function's behavior within the unit circle is crucial. The unit circle, a circle with a radius of 1 centered at the origin of the Cartesian plane, serves as a visual and conceptual tool for understanding trigonometric functions. The cosine of an angle, denoted as cos θ, is defined as the x-coordinate of the point where the terminal side of the angle intersects the unit circle. This definition immediately connects angles with their corresponding cosine values, laying the foundation for understanding equivalencies. Think of it as a map where each angle has a specific location (x-coordinate) that defines its cosine value. This x-coordinate varies as we move around the unit circle, and its behavior dictates the cosine function's properties.

Quadrantal Behavior of Cosine

The unit circle is divided into four quadrants, each spanning 90 degrees, and the cosine function exhibits distinct behavior in each quadrant. In the first quadrant (0° to 90°), cosine values are positive and range from 1 to 0. As we move into the second quadrant (90° to 180°), the x-coordinates become negative, hence cosine values are negative, ranging from 0 to -1. The third quadrant (180° to 270°) continues the trend of negative cosine values, ranging from -1 to 0. Finally, in the fourth quadrant (270° to 360°), cosine values revert to being positive, ranging from 0 to 1. Understanding this quadrantal behavior is essential for determining the sign of the cosine value for any given angle. For instance, since 120° lies in the second quadrant, we immediately know that cos 120° will be negative. This knowledge is a cornerstone for identifying equivalent expressions, as it narrows down the possibilities and helps in eliminating incorrect options. The symmetry of the unit circle also plays a significant role, which we'll explore further in the context of reference angles.

Determining cos 120° and Reference Angles

To find the value of cos 120°, we utilize the concept of reference angles. A reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. It essentially provides a way to relate trigonometric values of angles in different quadrants to those in the first quadrant, where trigonometric values are generally well-known. For 120°, which lies in the second quadrant, the reference angle is calculated by subtracting it from 180°: 180° - 120° = 60°. This means that the absolute value of cos 120° is the same as the absolute value of cos 60°, a familiar angle in the first quadrant.

Cosine of the Reference Angle

We know that cos 60° = 1/2. However, since 120° is in the second quadrant where cosine values are negative, we have cos 120° = -1/2. This understanding is critical because it allows us to connect the given angle to its reference angle and determine its exact cosine value, including the correct sign. Now, with the value of cos 120° firmly established, we can evaluate the given options to see which ones produce the same cosine value. Each option presents a different angle, and our task is to find the angle whose cosine matches -1/2. This process involves applying the same principles of reference angles and quadrantal behavior to each option, ensuring that we account for the sign and magnitude of the cosine value in the relevant quadrant. By methodically analyzing each choice, we can pinpoint the correct equivalent expression, solidifying our grasp of cosine function equivalencies.

Evaluating the Options

Now, let's delve into the options provided and determine which expression is equivalent to cos 120°, which we know equals -1/2. We will meticulously examine each option, applying the principles of reference angles and quadrantal behavior to ascertain their cosine values.

Option 1: cos 60°

The first option is cos 60°. As we previously established, 60° is a special angle in the first quadrant, and its cosine value is well-known. cos 60° = 1/2. Comparing this to our target value of -1/2, we can immediately see that cos 60° is not equivalent to cos 120°. The magnitudes are the same, but the signs differ, and this difference is crucial. In trigonometry, the sign of a trigonometric function depends on the quadrant in which the angle lies, and this distinction cannot be overlooked. Therefore, we can confidently eliminate this option.

Option 2: cos 240°

The second option is cos 240°. To evaluate this, we need to determine the reference angle for 240°. Since 240° lies in the third quadrant, we subtract 180° from it to find the reference angle: 240° - 180° = 60°. The reference angle is 60°, meaning the absolute value of cos 240° is the same as cos 60°, which is 1/2. However, in the third quadrant, cosine values are negative. Therefore, cos 240° = -1/2. This exactly matches our target value of cos 120°, making this option a strong contender for the correct answer. We will keep this option in mind as we proceed to analyze the remaining choices.

Option 3: cos 300°

The third option is cos 300°. The angle 300° lies in the fourth quadrant. To find the reference angle, we subtract 300° from 360°: 360° - 300° = 60°. Again, the reference angle is 60°, so the absolute value of cos 300° is the same as cos 60°, which is 1/2. In the fourth quadrant, cosine values are positive. Therefore, cos 300° = 1/2. This value does not match our target of -1/2, so we can eliminate this option.

Option 4: cos 420°

The final option is cos 420°. Since 420° is greater than 360°, it represents more than one full rotation around the unit circle. To find the equivalent angle within one rotation, we subtract 360°: 420° - 360° = 60°. Thus, cos 420° = cos 60° = 1/2. Once again, this value does not match our target of -1/2, allowing us to eliminate this option as well.

Final Answer: cos 240°

After meticulously evaluating each option, we can confidently conclude that the expression equivalent to cos 120° is cos 240°. Our analysis involved understanding the cosine function's behavior within the unit circle, utilizing reference angles, and carefully considering the sign of the cosine value in each quadrant. By calculating the cosine value for each option, we definitively showed that cos 240° is the only expression that yields the same value as cos 120°, which is -1/2. This process reinforces the fundamental principles of trigonometry and highlights the importance of understanding the properties of trigonometric functions and their relationships across different angles.

In conclusion, the question "Which expression is equivalent to cos 120°?" serves as an excellent exercise in applying trigonometric principles. Through a systematic approach involving the unit circle, reference angles, and quadrantal analysis, we determined that cos 240° is indeed equivalent to cos 120°. This journey underscores the significance of a solid foundation in trigonometry, especially the behavior of trigonometric functions across different quadrants. Understanding these concepts not only helps in solving specific problems but also builds a broader mathematical acumen. The ability to dissect trigonometric problems, apply relevant principles, and arrive at accurate conclusions is a valuable skill. As you continue your exploration of mathematics, remember that each problem solved is a step towards a deeper understanding of the subject and its applications. The concepts explored here are not just isolated facts; they are building blocks for more advanced topics in mathematics, physics, and engineering. Keep practicing, keep exploring, and the world of mathematics will continue to unfold its beauty and utility before you.