Finding Angles With Cosine Of 1/2 A Comprehensive Guide

#title: Finding Angles with Cosine of 1/2 A Comprehensive Guide

#repair-input-keyword: Determine all angle measures θ for which cos(θ) = 1/2 from the options: -120°, -60°, 120°, 600°, 660°.

Introduction: Understanding Cosine and Angles

In trigonometry, the cosine function relates an angle of a right triangle to the ratio of the adjacent side to the hypotenuse. More generally, in the unit circle, the cosine of an angle is represented by the x-coordinate of the point where the terminal side of the angle intersects the circle. Understanding this fundamental concept is crucial for solving trigonometric equations and exploring the periodic nature of trigonometric functions. In this article, we will delve into finding all angle measures for which the cosine function equals 1/2, focusing on the given options: -120°, -60°, 120°, 600°, and 660°. This exploration will not only help in identifying the correct answers but also in reinforcing the understanding of cosine values in different quadrants and the concept of coterminal angles. Grasping these principles will empower you to tackle a wide range of trigonometric problems with confidence and precision. This includes recognizing the symmetry and periodicity inherent in trigonometric functions, allowing for efficient problem-solving and a deeper appreciation of the mathematical relationships at play. So, let's embark on this journey to unveil the angles that satisfy the given condition, cos θ = 1/2, and solidify our grasp on the fundamentals of trigonometry.

The Unit Circle and Cosine Values

The unit circle is an invaluable tool in trigonometry. It's a circle with a radius of 1 centered at the origin of a coordinate plane. Angles are measured counterclockwise from the positive x-axis. The x-coordinate of a point on the unit circle corresponds to the cosine of the angle, while the y-coordinate corresponds to the sine of the angle. This visual representation makes it easier to understand the values of trigonometric functions for various angles. For instance, angles in the first and fourth quadrants have positive cosine values, while angles in the second and third quadrants have negative cosine values. The reference angle, which is the acute angle formed by the terminal side of the angle and the x-axis, plays a critical role in determining the trigonometric values. By understanding the symmetry of the unit circle and the reference angles, we can efficiently find angles with specific cosine values. This approach is particularly useful when dealing with angles outside the range of 0° to 360°, as we can find coterminal angles within this range to simplify the analysis. In our quest to find angles where cos θ = 1/2, the unit circle will be our guiding map, helping us pinpoint the exact locations where this condition is met. So, let's visualize the unit circle and prepare to navigate through its angles to solve our trigonometric puzzle.

Evaluating the Given Angle Measures

Now, let's evaluate each of the given angle measures to determine if their cosine is equal to 1/2. This involves understanding the concept of reference angles and the quadrants in which these angles lie. We will also consider coterminal angles, which are angles that share the same terminal side.

-120°

An angle of -120° is measured clockwise from the positive x-axis and lies in the third quadrant. The reference angle for -120° is 60° (180° - 120°). In the third quadrant, cosine values are negative. Since cos(60°) = 1/2, then cos(-120°) = -1/2. Therefore, -120° is not a solution.

-60°

An angle of -60° is measured clockwise from the positive x-axis and lies in the fourth quadrant. The reference angle for -60° is 60°. In the fourth quadrant, cosine values are positive. Since cos(60°) = 1/2, then cos(-60°) = 1/2. Therefore, -60° is a solution.

120°

An angle of 120° lies in the second quadrant. The reference angle for 120° is 60° (180° - 120°). In the second quadrant, cosine values are negative. Since cos(60°) = 1/2, then cos(120°) = -1/2. Therefore, 120° is not a solution.

600°

To find the cosine of 600°, we first find a coterminal angle within the range of 0° to 360°. We subtract 360° from 600° to get 240°. An angle of 240° lies in the third quadrant. The reference angle for 240° is 60° (240° - 180°). In the third quadrant, cosine values are negative. Since cos(60°) = 1/2, then cos(600°) = cos(240°) = -1/2. Therefore, 600° is not a solution.

660°

To find the cosine of 660°, we first find a coterminal angle within the range of 0° to 360°. We subtract 360° from 660° to get 300°. An angle of 300° lies in the fourth quadrant. The reference angle for 300° is 60° (360° - 300°). In the fourth quadrant, cosine values are positive. Since cos(60°) = 1/2, then cos(660°) = cos(300°) = 1/2. Therefore, 660° is a solution.

Identifying the Solutions: -60° and 660°

After evaluating each angle measure, we have identified that -60° and 660° are the angles for which cos(θ) = 1/2. These angles, when visualized on the unit circle, correspond to points where the x-coordinate is 1/2. This exercise underscores the importance of understanding the unit circle, reference angles, and the signs of trigonometric functions in different quadrants. It's crucial to remember that cosine is positive in the first and fourth quadrants, which is why -60° (fourth quadrant) and 660° (coterminal with 300° in the fourth quadrant) satisfy the equation. This methodical approach to evaluating each option not only provides the correct answers but also reinforces the fundamental concepts of trigonometry. Recognizing the relationship between angles and their cosine values is a cornerstone of trigonometric problem-solving. Therefore, mastering these concepts will pave the way for tackling more complex trigonometric challenges with confidence and precision. Keep practicing and exploring different angles to solidify your understanding of the cosine function and its behavior in the unit circle.

General Solutions and Periodicity

While we have identified specific solutions within the given options, it's important to understand the general solutions for the equation cos(θ) = 1/2. Due to the periodic nature of the cosine function, there are infinitely many angles that satisfy this equation. The cosine function has a period of 360°, which means that cos(θ) = cos(θ + 360k) for any integer k. Therefore, if θ₀ is a solution to cos(θ) = 1/2, then θ₀ + 360k is also a solution. In our case, we found that -60° and 660° are solutions. We can express the general solutions as:

• θ = -60° + 360k
• θ = 60° + 360k

Where k is any integer. These general solutions capture all possible angles that have a cosine of 1/2. Understanding the periodicity of trigonometric functions is essential for finding all solutions to trigonometric equations. By expressing the solutions in this general form, we acknowledge the infinite nature of the solutions and provide a complete picture of the angle measures that satisfy the given condition. This concept of periodicity is not only applicable to cosine but also to other trigonometric functions like sine and tangent. Grasping this principle allows for a more comprehensive understanding of trigonometric behavior and enables the solution of a wider range of problems. So, remember that when solving trigonometric equations, it's crucial to consider the general solutions and the periodic nature of the functions involved.

Conclusion: Mastering Trigonometric Concepts

In conclusion, we have successfully identified the angle measures for which cos(θ) = 1/2 from the given options. We determined that -60° and 660° are the solutions by leveraging our understanding of the unit circle, reference angles, and the signs of cosine in different quadrants. Furthermore, we explored the concept of general solutions and the periodicity of the cosine function, highlighting the infinite nature of solutions to trigonometric equations. This journey through the realm of trigonometry underscores the importance of a solid foundation in fundamental concepts. Mastering the unit circle, reference angles, and the behavior of trigonometric functions is crucial for solving a wide range of problems. By practicing and applying these concepts, you can develop a deeper understanding of trigonometry and enhance your problem-solving skills. Remember that trigonometry is not just about memorizing formulas; it's about understanding the relationships between angles and their trigonometric values. So, continue to explore, practice, and challenge yourself to unlock the full potential of your trigonometric knowledge. This will not only help you excel in your studies but also appreciate the beauty and elegance of mathematics in the world around us. Keep up the great work and keep exploring the fascinating world of trigonometry!