Unit Circle And Trigonometric Functions A Comprehensive Guide

(1.1) Decoding Terminal Points P(x, y) on the Unit Circle

In the realm of trigonometry, the unit circle serves as a cornerstone for understanding trigonometric functions and their relationships. A crucial aspect of working with the unit circle involves identifying terminal points, which are the points where the terminal side of an angle, measured in radians, intersects the circle. This section delves into the process of finding these pivotal points for various values of 't', representing the angle in radians. Grasping the concept of terminal points is fundamental, as they directly correspond to the cosine and sine values of the angle, forming the basis for understanding trigonometric functions.

When navigating the unit circle, it's essential to remember that a full revolution corresponds to 2π radians. Therefore, angles that differ by multiples of 2π are coterminal, meaning they share the same terminal point. This understanding is key to simplifying the process of finding terminal points for angles larger than 2π or negative angles. To determine the terminal point P(x, y) for a given value of t, we first need to find the coterminal angle within the range of 0 to 2π. This is achieved by adding or subtracting multiples of 2π until the angle falls within this range. Once we have the coterminal angle, we can identify the corresponding point on the unit circle using our knowledge of special angles and their associated coordinates. Special angles, such as π/6, π/4, and π/3, have well-defined coordinates on the unit circle, making it easier to determine the terminal points for their multiples and related angles.

To illustrate this concept, let's consider the angle t = -4π/3. Since this angle is negative, we add multiples of 2π until we obtain a positive coterminal angle within the desired range. Adding 2π to -4π/3 gives us 2π/3, which lies between 0 and 2π. Now, we can identify the terminal point for 2π/3 on the unit circle. Recognizing that 2π/3 is in the second quadrant and related to the special angle π/3, we can determine its coordinates. The x-coordinate will be negative, and the y-coordinate will be positive. Consulting the unit circle or our knowledge of trigonometric values, we find that the terminal point for 2π/3 is (-1/2, √3/2). Therefore, the terminal point for t = -4π/3 is also (-1/2, √3/2). This example highlights the importance of finding coterminal angles and utilizing our knowledge of special angles to efficiently determine terminal points on the unit circle. The ability to quickly and accurately identify terminal points is crucial for solving a wide range of trigonometric problems and understanding the behavior of trigonometric functions.

(a) t = -4π/3

As discussed in the general explanation, to find the terminal point P(x, y) on the unit circle determined by t = -4π/3, we first find a coterminal angle within the range of 0 to 2π. Adding 2π (which is 6π/3) to -4π/3 yields:

-4π/3 + 6π/3 = 2π/3

Thus, 2π/3 is a coterminal angle. Now, we need to identify the point on the unit circle corresponding to 2π/3. This angle lies in the second quadrant. We know that the reference angle for 2π/3 is π/3 (60 degrees). The coordinates for π/3 in the first quadrant are (1/2, √3/2). In the second quadrant, the x-coordinate is negative, and the y-coordinate is positive. Therefore, the terminal point P(x, y) for t = -4π/3 is:

P(x, y) = (-1/2, √3/2)

(b) t = 8π/3

For t = 8π/3, we again need to find a coterminal angle between 0 and 2π. Subtracting 2π (which is 6π/3) from 8π/3 gives us:

8π/3 - 6π/3 = 2π/3

This is the same coterminal angle as in part (a), so the terminal point P(x, y) is the same:

P(x, y) = (-1/2, √3/2)

(c) t = 11π/6

The angle t = 11π/6 is already within the range of 0 to 2π. This angle lies in the fourth quadrant. The reference angle for 11π/6 is 2π - 11π/6 = π/6 (30 degrees). The coordinates for π/6 in the first quadrant are (√3/2, 1/2). In the fourth quadrant, the x-coordinate is positive, and the y-coordinate is negative. Therefore, the terminal point P(x, y) for t = 11π/6 is:

P(x, y) = (√3/2, -1/2)

(d) t = 3π/2

The angle t = 3π/2 is a quadrantal angle, meaning it lies on an axis. Specifically, it lies on the negative y-axis. The coordinates for this point on the unit circle are:

P(x, y) = (0, -1)

(1.2) Unveiling Trigonometric Values: Sine, Cosine, and Tangent

Suppose (5/13, -12/13) is a point on the unit circle corresponding to an angle t. The fundamental connection between points on the unit circle and trigonometric functions lies in the fact that the coordinates of these points directly represent the cosine and sine values of the angle. Specifically, the x-coordinate corresponds to the cosine of the angle, and the y-coordinate corresponds to the sine of the angle. Therefore, if (5/13, -12/13) is the terminal point for angle t, we can immediately deduce that cos(t) = 5/13 and sin(t) = -12/13. This relationship forms the cornerstone for understanding and evaluating trigonometric functions.

Building upon the sine and cosine values, we can determine the remaining trigonometric functions, such as tangent, cotangent, secant, and cosecant. The tangent function, denoted as tan(t), is defined as the ratio of sine to cosine, that is, tan(t) = sin(t) / cos(t). Given the values of sin(t) and cos(t), we can readily calculate tan(t). In this case, tan(t) = (-12/13) / (5/13) = -12/5. Similarly, the cotangent function, denoted as cot(t), is the reciprocal of the tangent function, or cot(t) = 1 / tan(t). Therefore, cot(t) = -5/12. The secant function, denoted as sec(t), is the reciprocal of the cosine function, sec(t) = 1 / cos(t). Hence, sec(t) = 13/5. Finally, the cosecant function, denoted as csc(t), is the reciprocal of the sine function, csc(t) = 1 / sin(t). Thus, csc(t) = -13/12. By understanding the definitions and relationships between these trigonometric functions, we can efficiently calculate their values given the coordinates of a point on the unit circle.

It's worth noting the sign conventions for trigonometric functions in different quadrants of the unit circle. In the first quadrant, all trigonometric functions are positive. In the second quadrant, only sine and cosecant are positive. In the third quadrant, only tangent and cotangent are positive. In the fourth quadrant, only cosine and secant are positive. These sign conventions are crucial for determining the correct sign of trigonometric function values based on the quadrant in which the angle lies. In this example, the point (5/13, -12/13) lies in the fourth quadrant, where cosine is positive and sine is negative, which aligns with our calculated values. Mastering the relationship between points on the unit circle, trigonometric functions, and their signs is essential for success in trigonometry and related fields.

Therefore:

• sin(t) = -12/13
• cos(t) = 5/13

Using these values, we can find the remaining trigonometric functions:

• tan(t) = sin(t) / cos(t) = (-12/13) / (5/13) = -12/5
• cot(t) = 1 / tan(t) = -5/12
• sec(t) = 1 / cos(t) = 13/5
• csc(t) = 1 / sin(t) = -13/12