Problem 25 - Algebra and Trigonometry

Question

\(21-30=\) Find the terminal point \(P(x, y)\) on the unit circle determined by the given value of \(t .\) $$ t=-\frac{\pi}{3} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

We need to find the terminal point on the unit circle corresponding to the angle \( t = -\frac{\pi}{3} \). In the context of a unit circle, the angle is measured in radians from the positive x-axis counterclockwise. A negative angle means we measure clockwise.

Step 2: Locating the Angle on the Unit Circle

The angle \( -\frac{\pi}{3} \) means we need to start from the positive x-axis and move clockwise \( \frac{\pi}{3} \) radians. The equivalent positive angle is \( 2\pi - \frac{\pi}{3} = \frac{5\pi}{3} \). This places the angle -\(\frac{\pi}{3}\) in the fourth quadrant of the unit circle.

Step 3: Determining Coordinates on the Unit Circle

In the unit circle, the coordinates of any angle \( \theta \) are given by \((\cos(\theta), \sin(\theta))\). For \( \theta = -\frac{\pi}{3} \), we calculate: \( \cos\left(-\frac{\pi}{3}\right) = \frac{1}{2} \) and \( \sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2} \).

Step 4: Conclusion

Hence, the terminal point \( P(x, y) \) is found on the unit circle at \( \left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right) \). These are the coordinates corresponding to the angle \(-\frac{\pi}{3}\) on the unit circle.

Key Concepts

Terminal Point

On the unit circle, a terminal point corresponds to an angle measured from the positive x-axis. When you have an angle, whether positive or negative, it helps locate a specific point on the circle's perimeter. Understanding terminal points is crucial as it provides a visual representation of trigonometric functions.

• The unit circle has a radius of 1, centralizing all points on the perimeter at this distance from the origin.
• Terminal points can be identified using the angle, typically marked as \( t \) or \( \theta \).

A negative angle changes the direction of measurement. Instead of moving counterclockwise as with a positive angle, you would measure clockwise. The terminal point's coordinates are determined by the cosine and sine values of the angle in question. Therefore, discovering a terminal point is about translating angular measurement into a coordinate position.

Radian Measure

Radians are a way of measuring angles based on the unit circle. Unlike degrees, radians provide a natural connection between the angle measure and the arc length. On the unit circle, the full circumference measures \(2\pi\) radians.When an angle is negative, as in \( -\frac{\pi}{3} \), it indicates a clockwise traversal. Such an angle means starting from the positive x-axis and heading backwards. This measure corresponds to the same location as if you moved \( 2\pi - \frac{\pi}{3} = \frac{5\pi}{3} \) radians counterclockwise.Remember:

• One complete rotation is \(2\pi\) radians, equivalent to 360 degrees.
• An angle of \( \pi \) radians equals 180 degrees.
• \( \frac{\pi}{2} \) radians equals 90 degrees.

Understanding radian measure helps with visualizing the angle's actual placement around the circle and is conducive to calculating coordinates on the unit circle.

Trigonometric Coordinates

Trigonometric coordinates on the unit circle mean assigning values to the point based on trigonometric functions. The essence of this is to use the circle's angle to find corresponding \((x, y)\) coordinates through sine and cosine functions.

• The x-coordinate of a terminal point is given by \(\cos(t)\).
• The y-coordinate is given by \(\sin(t)\).

For \( t = -\frac{\pi}{3} \), - The cosine function provides \( \cos\left(-\frac{\pi}{3}\right) = \frac{1}{2} \).- The sine function results in \( \sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2} \).These calculations yield the trigonometric coordinates \( (\frac{1}{2}, -\frac{\sqrt{3}}{2}) \).Understanding trigonometric coordinates is helpful in solving various problems related to angles, such as finding points or analyzing motion, as they provide a mathematical method to describe position on the unit circle.