Problem 27 - Algebra and Trigonometry

Question

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

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 \(P(x, y)\) on the unit circle corresponding to the angle \(t = -\frac{\pi}{3}\). The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. The angle \(t\) is measured from the positive x-axis, counterclockwise for positive angles and clockwise for negative angles.

Step 2: Locating the Angle on the Unit Circle

The angle \(-\frac{\pi}{3}\) indicates a negative direction, meaning we move \(\frac{\pi}{3}\) radians clockwise from the positive x-axis. This places us in the fourth quadrant of the unit circle, as \(\frac{\pi}{3}\) radians is equivalent to 60 degrees.

Step 3: Finding the Coordinates of the Terminal Point

In the unit circle, the coordinates of any point \(P(x, y)\) are given by \((\cos t, \sin t)\). For \(t = -\frac{\pi}{3}\), using trigonometric identities, we find: \[ \cos\left(-\frac{\pi}{3}\right) = \cos\frac{\pi}{3} = \frac{1}{2} \] \[ \sin\left(-\frac{\pi}{3}\right) = -\sin\frac{\pi}{3} = -\frac{\sqrt{3}}{2} \].

Step 4: Writing the Final Result

Thus, the coordinates of the terminal point \(P(x, y)\) for \(t = -\frac{\pi}{3}\) on the unit circle are \(\left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right)\).

Key Concepts

Understanding Trigonometric Identities

Trigonometric identities are mathematical equations that relate different trigonometric functions to each other. In this exercise, we use these identities to find coordinates on the unit circle. Here are some key points:

• The most common trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)).
• On the unit circle, the cosine of an angle represents the x-coordinate, while the sine of an angle represents the y-coordinate.

In the problem, we use the identities to evaluate trigonometric functions for the angle \(-\frac{\pi}{3}\):

• \( \cos(-\frac{\pi}{3}) = \cos(\frac{\pi}{3}) = \frac{1}{2} \)
• \( \sin(-\frac{\pi}{3}) = -\sin(\frac{\pi}{3}) = -\frac{\sqrt{3}}{2} \)

These identities help us find the terminal point by simply evaluating the known values for angles on the unit circle.

Locating the Terminal Point on the Unit Circle

The terminal point refers to the specific point on the unit circle corresponding to a particular angle. The unit circle is centered at the origin (0,0) and has a radius of 1. The position of a terminal point is crucial in understanding trigonometric functions.

• An angle in standard position starts on the positive x-axis.
• A positive angle is measured counterclockwise from the positive x-axis, while a negative angle is measured clockwise.
• In our example, the angle \(-\frac{\pi}{3}\) is measured in a clockwise direction, placing the terminal point in the fourth quadrant of the unit circle.

Identifying the quadrant is useful, as it tells us the signs of the sine and cosine functions. In the fourth quadrant, cosine is positive and sine is negative, which corresponds to the coordinates \( \left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right) \).

Basics of Angle Measurement

Angle measurement is fundamental to understanding trigonometry and involves determining how far you've rotated around a circle from a starting point.

• Angles on the unit circle can be measured in degrees or radians, with one full circle equivalent to 360 degrees or \(2\pi\) radians.
• In this exercise, we use radians, which are favored in mathematics because they provide a natural connection to the arc length and radius.
• An angle of \(-\frac{\pi}{3}\) radians means rotating clockwise \( \frac{\pi}{3} \) radians, equivalent to 60 degrees.

Radian measure simplifies many calculations and connects directly to the properties of the circle, making it the standard in many mathematical contexts.