Equivalent Trigonometric Expressions For Cos 120 Degrees

In the realm of trigonometry, understanding the relationships between angles and their trigonometric functions is crucial. This article delves into the question: Which expression is equivalent to cos 120°? We'll explore the properties of the cosine function, its behavior in different quadrants, and how to use reference angles to find equivalent expressions. We will analyze the provided options, which include cos 60°, cos 240°, cos 300°, and cos 420°, and determine which one matches the value of cos 120°.

Decoding the Cosine Function

To determine which expression is equivalent to cos 120°, we must first understand the cosine function and its properties. The cosine function, often abbreviated as cos, is one of the fundamental trigonometric functions. In a right-angled triangle, the cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse. However, when dealing with angles beyond the range of 0° to 90°, we use the unit circle definition. The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. For any angle θ, the cosine of θ is the x-coordinate of the point where the terminal side of the angle intersects the unit circle.

Understanding the Unit Circle is pivotal in grasping the behavior of trigonometric functions across all quadrants. The unit circle is divided into four quadrants, each spanning 90 degrees. In the first quadrant (0° to 90°), both the x and y coordinates are positive, meaning that both cosine and sine are positive. In the second quadrant (90° to 180°), the x-coordinate is negative while the y-coordinate is positive, making cosine negative and sine positive. In the third quadrant (180° to 270°), both x and y coordinates are negative, so both cosine and sine are negative. Finally, in the fourth quadrant (270° to 360°), the x-coordinate is positive and the y-coordinate is negative, meaning cosine is positive and sine is negative. This cyclical pattern is essential for determining the sign of trigonometric functions for any angle.

The cosine function is periodic, which means its values repeat after a certain interval. The period of the cosine function is 360°, meaning that cos(θ) = cos(θ + 360°k) for any integer k. This property allows us to find equivalent expressions for cosine by adding or subtracting multiples of 360° from the angle. Furthermore, the cosine function is an even function, which means that cos(-θ) = cos(θ). This symmetry is another crucial aspect to consider when finding equivalent expressions.

Evaluating cos 120°

The question at hand is to find an expression equivalent to cos 120°. The angle 120° lies in the second quadrant. As mentioned earlier, cosine is negative in the second quadrant. To find the reference angle for 120°, we subtract it from 180°: 180° - 120° = 60°. The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. The cosine of 120° will have the same magnitude as the cosine of its reference angle, 60°, but with a negative sign because it is in the second quadrant. Therefore, cos 120° = -cos 60°.

To determine the exact value, recall the values of trigonometric functions for special angles. The cosine of 60° is a well-known value: cos 60° = 1/2. Therefore, cos 120° = -cos 60° = -1/2. This value serves as our benchmark for evaluating the given options. Understanding this foundational concept is key to correctly identifying equivalent expressions. By breaking down the problem into smaller, manageable steps, we can methodically analyze each option and arrive at the correct solution.

Analyzing the Options

Now that we know cos 120° = -1/2, let's examine each of the provided options to see which one yields the same value. This process involves understanding how the cosine function behaves in different quadrants and utilizing reference angles.

Option 1: cos 60°

The first option is cos 60°. As we've already established, cos 60° = 1/2. This is a positive value, while cos 120° is -1/2. Therefore, cos 60° is not equivalent to cos 120°. This option serves as a crucial reference point, highlighting the importance of considering the sign of the cosine function in different quadrants.

Option 2: cos 240°

The second option is cos 240°. The angle 240° lies in the third quadrant, where cosine is negative. To find the reference angle for 240°, we subtract 180° from it: 240° - 180° = 60°. The reference angle is 60°. Thus, cos 240° = -cos 60° = -1/2. This matches the value of cos 120°, making it a potential equivalent expression.

Option 3: cos 300°

The third option is cos 300°. The angle 300° lies in the fourth quadrant, where cosine is positive. To find the reference angle for 300°, we subtract it from 360°: 360° - 300° = 60°. The reference angle is 60°. Therefore, cos 300° = cos 60° = 1/2. This is not equal to cos 120°, so this option is incorrect.

Option 4: cos 420°

The fourth option is cos 420°. Since the cosine function has a period of 360°, we can subtract 360° from 420° to find a coterminal angle: 420° - 360° = 60°. Thus, cos 420° = cos 60° = 1/2. This value does not match cos 120°, so this option is also incorrect.

Conclusion: The Equivalent Expression

After analyzing all the options, we can confidently conclude that the expression equivalent to cos 120° is cos 240°. We arrived at this conclusion by understanding the properties of the cosine function, its behavior in different quadrants, and the concept of reference angles. We calculated that cos 120° = -1/2 and found that cos 240° also equals -1/2.

This exercise highlights the importance of a solid foundation in trigonometric principles. By understanding the unit circle, reference angles, and the periodicity of trigonometric functions, one can confidently navigate and solve problems involving trigonometric identities and equivalent expressions. The ability to manipulate and simplify trigonometric expressions is a crucial skill in various fields, including mathematics, physics, and engineering. Therefore, mastering these concepts is an investment in one's academic and professional future. Ultimately, breaking down complex problems into manageable steps and applying fundamental principles will lead to success in trigonometry and beyond. This exploration of equivalent trigonometric expressions underscores the beauty and interconnectedness of mathematical concepts.