Finding Terminal Points On The Unit Circle P(x, Y) For Given T Values

This article delves into the process of pinpointing terminal points P(x, y) on the unit circle, a fundamental concept in trigonometry and precalculus. We will explore how to determine these points for specific values of t, which represent the arc length along the unit circle, starting from the positive x-axis. Understanding this concept is crucial for grasping trigonometric functions, their periodicity, and their relationship to the unit circle. This comprehensive guide aims to provide a clear and detailed explanation, making it accessible for students and enthusiasts alike. Let's embark on this journey to unravel the intricacies of the unit circle and terminal points, paving the way for a deeper understanding of trigonometry.

(a) t = -4π/3

To find the terminal point P(x, y) on the unit circle for t = -4π/3, we need to understand how angles and arc lengths relate on the unit circle. The unit circle, with a radius of 1, serves as a visual tool for understanding trigonometric functions. The angle t represents the arc length along the circumference of the circle, measured counterclockwise from the positive x-axis. A negative value of t, like our case of t = -4π/3, signifies that we move in a clockwise direction along the circle.

First, let's consider the angle -4π/3. Since a full circle is 2π, we can add 2π to -4π/3 to find a coterminal angle (an angle that shares the same terminal point) within the range of 0 to 2π. Adding 2π (which is 6π/3) to -4π/3 gives us 2π/3. This means that the angle -4π/3 is coterminal with the angle 2π/3. The coterminal angle simplifies our task because we can now work with an angle in the standard range of 0 to 2π.

The angle 2π/3 lies in the second quadrant of the unit circle. To determine the coordinates of the terminal point, we need to find the reference angle. The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. In the second quadrant, the reference angle is calculated by subtracting the angle from π. Thus, the reference angle for 2π/3 is π - 2π/3 = π/3. The reference angle, π/3, is a special angle whose trigonometric values are well-known. It corresponds to 60 degrees.

Now, we can use the reference angle to find the coordinates of the terminal point. Recall that on the unit circle, the x-coordinate corresponds to the cosine of the angle, and the y-coordinate corresponds to the sine of the angle. For the reference angle π/3, we know that cos(π/3) = 1/2 and sin(π/3) = √3/2. However, since the angle 2π/3 lies in the second quadrant, the x-coordinate will be negative, and the y-coordinate will be positive. Therefore, the coordinates of the terminal point for t = -4π/3 are (-1/2, √3/2). This detailed process highlights the importance of understanding coterminal angles, reference angles, and the relationship between angles and coordinates on the unit circle.

(b) t = 8π/3

Moving on to the next value, t = 8π/3, we again seek to find the terminal point P(x, y) on the unit circle. The approach remains similar: we first find a coterminal angle within the range of 0 to 2π, then determine the reference angle, and finally, use the reference angle to identify the coordinates of the terminal point. The systematic approach ensures accuracy and a clear understanding of the underlying principles.

The given angle, 8π/3, is larger than 2π. To find a coterminal angle, we subtract multiples of 2π until we obtain an angle within the desired range. Subtracting 2π (or 6π/3) from 8π/3 gives us 2π/3. This coterminal angle, 2π/3, is the same as the one we encountered in part (a). This immediately tells us that the terminal point will be in the second quadrant, further emphasizing the cyclical nature of trigonometric functions and the importance of identifying coterminal angles.

As we already determined in part (a), the angle 2π/3 lies in the second quadrant, and its reference angle is π/3. The cosine and sine values for the reference angle π/3 are 1/2 and √3/2, respectively. Considering the quadrant, the x-coordinate will be negative, and the y-coordinate will be positive. Consequently, the terminal point P(x, y) for t = 8π/3 is (-1/2, √3/2). The fact that we arrived at the same terminal point as in part (a), despite the different initial values of t, underscores the periodic nature of trigonometric functions. This periodicity is a key characteristic of trigonometric functions, and understanding it is crucial for solving a wide range of problems.

(c) t = 11π/6

For t = 11π/6, we follow the same methodology to locate the corresponding terminal point P(x, y) on the unit circle. The goal is to break down the problem into manageable steps, utilizing the concepts of coterminal angles, reference angles, and the relationship between angles and coordinates on the unit circle. This step-by-step approach is vital for achieving clarity and accuracy in trigonometric calculations.

The angle 11π/6 is already within the range of 0 to 2π, so we don't need to find a coterminal angle. This simplifies our task, allowing us to directly focus on determining the quadrant in which the angle lies and calculating the reference angle. Recognizing that the angle is within the standard range is an important first step in efficiently solving the problem.

The angle 11π/6 lies in the fourth quadrant. To find the reference angle in the fourth quadrant, we subtract the angle from 2π. Therefore, the reference angle for 11π/6 is 2π - 11π/6 = π/6. The reference angle, π/6, is another special angle, corresponding to 30 degrees. The trigonometric values for this angle are well-known and crucial for determining the coordinates of the terminal point. Understanding the common trigonometric values for special angles is a fundamental skill in trigonometry.

For the reference angle π/6, we know that cos(π/6) = √3/2 and sin(π/6) = 1/2. In the fourth quadrant, the x-coordinate is positive, and the y-coordinate is negative. Therefore, the coordinates of the terminal point for t = 11π/6 are (√3/2, -1/2). The combination of understanding reference angles and quadrant signs allows us to accurately pinpoint the terminal point on the unit circle. This highlights the importance of visual understanding and applying the correct sign conventions.

(d) t = 3π

Finally, let's determine the terminal point P(x, y) for t = 3π. This case provides an opportunity to reinforce our understanding of coterminal angles and how they simplify the process of finding terminal points. The concept of coterminal angles is particularly useful when dealing with angles larger than 2π or negative angles.

The angle 3π is greater than 2π. To find a coterminal angle within the range of 0 to 2π, we subtract 2π from 3π, which gives us π. The coterminal angle π is a key angle on the unit circle, lying on the negative x-axis. Identifying the coterminal angle simplifies the problem considerably, as we can now work with a well-known angle.

The angle π corresponds to the point on the unit circle where the terminal side intersects the negative x-axis. At this point, the x-coordinate is -1, and the y-coordinate is 0. Therefore, the coordinates of the terminal point for t = 3π are (-1, 0). This case demonstrates how understanding special angles and their corresponding points on the unit circle can lead to quick and accurate solutions. The ability to recognize and work with special angles is an essential skill in trigonometry.

In conclusion, this exploration has provided a comprehensive guide to finding terminal points on the unit circle for various values of t. By systematically applying the concepts of coterminal angles, reference angles, and quadrant signs, we can accurately determine the coordinates of these points. This understanding forms a crucial foundation for further studies in trigonometry and related fields. The unit circle serves as a powerful visual tool, and mastering its intricacies is key to unlocking deeper understanding of trigonometric functions and their applications.