Problem 29 - Algebra and Trigonometry

Question

23-32 \(\approx\) Find the terminal point \(P(X, y)\) on the unit circle determined by the given value of \(t\) $$ t=\frac{2 \pi}{3} $$

Step-by-Step Solution

Answer

The terminal point is \( P = \left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right) \).

Step 1: Understanding the Problem

To find the terminal point on the unit circle for the value of \( t = \frac{2\pi}{3} \), we need to determine the coordinates \( (X, Y) \) corresponding to the angle \( t \). The unit circle has a radius of 1, and each point \( (X, Y) \) on it can be found using trigonometric functions.

Step 2: Evaluate the Cosine of the Angle

The x-coordinate of the terminal point is found by evaluating the cosine of the angle \( t = \frac{2\pi}{3} \). Cosine is an even function, and for angles in radians, the cosine \( \cos \left( \frac{2\pi}{3} \right) \) corresponds to the x-coordinate. Since \( \frac{2\pi}{3} \approx 120^{\circ} \) is in the second quadrant, where cosine values are negative, we get:\[ X = \cos \left( \frac{2\pi}{3} \right) = -\frac{1}{2} \]

Step 3: Evaluate the Sine of the Angle

The y-coordinate of the terminal point is found by evaluating the sine of the angle \( t = \frac{2\pi}{3} \). Sine is a periodic function, and for angles within the second quadrant, it retains positive values. Thus, for \( \sin \left( \frac{2\pi}{3} \right) \), we find:\[ Y = \sin \left( \frac{2\pi}{3} \right) = \frac{\sqrt{3}}{2} \]

Step 4: Write the Terminal Point

We now have the values for both coordinates based on the evaluations of cosine and sine. Therefore, the terminal point \( P(X, Y) \) on the unit circle when \( t = \frac{2\pi}{3} \) is:\[ P = \left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right) \]

Key Concepts

Trigonometric Functions

Trigonometric functions are foundational concepts in mathematics. They are especially important when dealing with angles and circles. In the context of the unit circle, trigonometric functions such as sine and cosine help to define the coordinates of points on this circle.

• Cosine Function: Gives the x-coordinate of a point on the unit circle. It is crucial in problems involving the unit circle since it helps determine the horizontal component of a point's position.


• Sine Function: Provides the y-coordinate of a point on the unit circle. This function is key in finding the vertical component of a point's position on the circle.

The unit circle is important because all points on it have a radius of 1. This simple but powerful characteristic allows these functions to consistently relate angles in radians to points on the circle. These functions cycle through their values as angles increase from 0 to 2π radians, repeating every full rotation (360°). Understanding how to find these values on the unit circle is immensely helpful for solving more complex problems with angles.

Terminal Point

On the unit circle, the terminal point is the final position of a point after tracing an angle from the positive x-axis. It is defined by the coordinates (X, Y) given by the cosine and sine of the angle, respectively.

• Determining the Quadrant: Where the terminal point lies depends on the angle. In our example, with an angle of \(t = \frac{2\pi}{3}\), the terminal point is in the second quadrant.


• Features of Quadrants: The second quadrant is where x-values (cosine) are negative, and y-values (sine) are positive. This is crucial for interpreting the trigonometric function outputs correctly.

Understanding the terminal point is essential because it ties together the angle and its representation on the unit circle through trigonometric functions. This helps in visualizing mathematics in spatial terms, providing a more intuitive grasp of how angles translate into positions.

Cosine and Sine

Cosine and sine are the key trigonometric functions used to find coordinates of points on the unit circle. These functions show the relationship between the angle and its effects on position.

• Cosine of \(\frac{2\pi}{3}\): The x-coordinate. For this angle, \(\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}\).


• Sine of \(\frac{2\pi}{3}\): The y-coordinate. For this angle, \(\sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}\).

These two functions give the specific terminal point \(P = \left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right)\) on the unit circle. The cosine tells us about the horizontal distance from the origin, while the sine describes the vertical distance. Their coordinated use allows for a comprehensive underlining of an angle's impact, making them indispensable tools in trigonometry.