Find The Exact Value Of Cos(-13π/6) A Step-by-Step Guide

In this comprehensive article, we will delve into the process of determining the exact value of the trigonometric function cos(-13π/6). This exploration will not only reinforce your understanding of trigonometric functions but also enhance your ability to navigate angles beyond the standard 0 to 2π range. We will begin by understanding the properties of cosine function, including its periodicity and symmetry, and then apply these properties to simplify the given angle. Subsequently, we will locate the equivalent angle within the unit circle, allowing us to easily determine the cosine value. This step-by-step approach will ensure that you grasp the underlying concepts and techniques, enabling you to confidently tackle similar trigonometric problems in the future. Whether you are a student seeking to master trigonometry or an enthusiast eager to expand your mathematical toolkit, this article provides a clear and concise guide to solving trigonometric functions.

The cosine function is a fundamental concept in trigonometry, representing the x-coordinate of a point on the unit circle corresponding to a given angle. Understanding its properties is crucial for evaluating trigonometric expressions. One of the key properties is its periodicity: the cosine function repeats its values every 2π radians. This means that cos(θ) = cos(θ + 2πk) for any integer k. This property allows us to simplify angles that are outside the standard range of 0 to 2π. Another important property is its symmetry: the cosine function is an even function, meaning that cos(-θ) = cos(θ). This property is helpful when dealing with negative angles, as it allows us to convert them to their positive counterparts. By understanding these properties, we can effectively manipulate and simplify trigonometric expressions, making it easier to find their exact values. The cosine function's behavior can be visualized on the unit circle, where the x-coordinate represents the cosine value for any given angle. As the angle rotates around the circle, the x-coordinate oscillates between -1 and 1, reflecting the cosine function's range. Mastering these fundamental aspects of the cosine function is essential for success in trigonometry and related fields.

Understanding reference angles and their relationship to trigonometric functions is crucial for finding exact values. A reference angle is the acute angle formed between the terminal side of a given angle and the x-axis. This angle provides a simplified way to determine the trigonometric values of angles in any quadrant. To find the reference angle, we first determine the quadrant in which the terminal side of the angle lies. Then, we use specific formulas based on the quadrant. For example, if the angle is in the second quadrant, the reference angle is found by subtracting the angle from π. In the third quadrant, we subtract π from the angle, and in the fourth quadrant, we subtract the angle from 2π. Once we have the reference angle, we can determine the trigonometric values of the original angle by considering the signs of the trigonometric functions in that quadrant. For instance, in the second quadrant, sine is positive, while cosine and tangent are negative. By using reference angles, we can simplify complex trigonometric problems and find exact values efficiently. This technique is particularly useful for angles that are multiples of π/6, π/4, and π/3, as their trigonometric values are well-known. Reference angles help us relate angles in different quadrants to the first quadrant, where the trigonometric values are most easily remembered.

To evaluate cos(-13π/6), we'll employ a systematic approach, leveraging the properties of cosine and reference angles. Our primary goal is to find an equivalent angle within the familiar range of 0 to 2π, simplifying the evaluation process. First, we acknowledge the periodicity of the cosine function. Since cosine has a period of 2π, we can add or subtract multiples of 2π without changing the function's value. This allows us to bring the angle -13π/6 into a more manageable range. We add 2π (or 12π/6) to -13π/6, resulting in -π/6. However, this angle is still negative, so we add another 2π (or 12π/6), which gives us 11π/6. Now we have an angle within the desired range of 0 to 2π that is coterminal with -13π/6. Next, we determine the reference angle for 11π/6. This angle lies in the fourth quadrant, where cosine is positive. The reference angle is calculated as 2π - 11π/6, which equals π/6. The cosine of π/6 is a well-known value, √3/2. Since cosine is positive in the fourth quadrant, the cosine of 11π/6 is also √3/2. Therefore, cos(-13π/6) = cos(11π/6) = √3/2. This step-by-step method demonstrates how we can simplify a complex angle using periodicity and reference angles to find the exact trigonometric value.

Step-by-Step Solution for cos(-13π/6)

To find the exact value of cos(-13π/6), we will follow a detailed, step-by-step approach. This method will not only provide the solution but also reinforce the fundamental concepts of trigonometry, including the periodicity and symmetry of the cosine function, as well as the use of reference angles. By breaking down the problem into smaller, manageable steps, we can ensure clarity and accuracy in our calculations. Each step will be explained in detail, making it easier to understand the reasoning behind each operation. This comprehensive approach will equip you with the skills to tackle similar trigonometric problems confidently. Whether you are a student learning trigonometry or someone looking to refresh your knowledge, this step-by-step solution will serve as a valuable guide.

Step 1: Simplify the Angle Using Periodicity

The first step in evaluating cos(-13π/6) is to simplify the angle. The cosine function has a period of 2π, meaning that cos(θ) = cos(θ + 2πk) for any integer k. This property allows us to add or subtract multiples of 2π to the angle without changing its cosine value. Our goal is to find an angle coterminal with -13π/6 that lies within the range of 0 to 2π. To do this, we can add multiples of 2π (which is 12π/6) to -13π/6 until we obtain an angle in this range. Adding 2π (12π/6) once gives us -13π/6 + 12π/6 = -π/6. This angle is still negative, so we add another 2π (12π/6), resulting in -π/6 + 12π/6 = 11π/6. Now we have an angle, 11π/6, which is coterminal with -13π/6 and lies within the 0 to 2π range. Therefore, cos(-13π/6) = cos(11π/6). This simplification is crucial because it brings the angle into a more familiar range, making it easier to determine its reference angle and cosine value. By using the periodicity of the cosine function, we've taken the first step towards finding the exact value.

Step 2: Determine the Reference Angle

The next crucial step involves determining the reference angle for 11π/6. The reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. This angle helps us relate the trigonometric values of angles in different quadrants to the values in the first quadrant, where they are more easily remembered. Since 11π/6 lies in the fourth quadrant (where angles are between 3π/2 and 2π), we find the reference angle by subtracting 11π/6 from 2π. This calculation is as follows: 2π - 11π/6 = 12π/6 - 11π/6 = π/6. Therefore, the reference angle for 11π/6 is π/6. Knowing the reference angle is essential because it allows us to use the known trigonometric values of π/6 to find the values for 11π/6. This step simplifies the problem by reducing it to finding the cosine of a more familiar angle. Understanding how to determine reference angles is a fundamental skill in trigonometry, enabling us to evaluate trigonometric functions for a wide range of angles.

Step 3: Find the Cosine of the Reference Angle

Having determined the reference angle as π/6, the next step is to find the cosine of this angle. The cosine of π/6 (or 30 degrees) is a well-known trigonometric value that is often memorized or readily available in trigonometric tables. Recall that π/6 corresponds to a special right triangle, specifically a 30-60-90 triangle. In this triangle, the ratio of the adjacent side to the hypotenuse for the 30-degree angle is √3/2. Therefore, cos(π/6) = √3/2. This value is a cornerstone of trigonometry, and its understanding is crucial for solving a variety of problems. By knowing the cosine of the reference angle, we are one step closer to finding the cosine of the original angle, cos(-13π/6). This step highlights the importance of memorizing or having quick access to the trigonometric values of common angles, such as π/6, π/4, and π/3.

Step 4: Determine the Sign of Cosine in the Quadrant

Now that we have the cosine of the reference angle, √3/2, the final step is to determine the sign of the cosine function in the quadrant where the original angle, 11π/6, lies. As established earlier, 11π/6 is located in the fourth quadrant. In the fourth quadrant, the x-coordinates are positive, and since cosine corresponds to the x-coordinate on the unit circle, cosine is positive in this quadrant. Therefore, cos(11π/6) will have the same sign as cos(π/6), which is positive. Since cos(π/6) = √3/2, and cosine is positive in the fourth quadrant, we can conclude that cos(11π/6) = √3/2. This step underscores the significance of understanding the signs of trigonometric functions in different quadrants. By considering the quadrant in which the angle lies, we can accurately determine the sign of the trigonometric value, ensuring the correct solution. This completes the process of finding the exact value of cos(-13π/6).

Final Answer

By following these steps, we have successfully determined the exact value of cos(-13π/6). We simplified the angle using periodicity, found the reference angle, determined the cosine of the reference angle, and considered the sign of cosine in the appropriate quadrant. Therefore, the exact value of cos(-13π/6) is √3/2. This result demonstrates the power of understanding trigonometric properties and applying them systematically to solve problems. The final answer is:

cos(-13π/6) = √3/2

This comprehensive solution not only provides the answer but also reinforces the underlying concepts and techniques, making it a valuable learning resource for anyone studying trigonometry.