Problem 43 - Precalculus Mathematics for Calculus

Question

Find (a) the reference number for each value of \(t\) and (b) the terminal point determined by \(t\). $$t=-\frac{2 \pi}{3}$$

Step-by-Step Solution

Answer

Reference number: \(\frac{\pi}{3}\); Terminal point: \(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\).

Step 1: Understanding Reference Numbers

First, let's understand what a reference number is. A reference number is the positive angle between the terminal side of a given angle and the x-axis. It helps us to simplify the process of finding trigonometric values for angles.

Step 2: Determine the Quadrant of t

The angle \(t = -\frac{2\pi}{3}\) is a negative angle, which means we measure it clockwise from the positive x-axis. Since \(-\frac{2\pi}{3}\) lies within the first negative revolution of the circle (i.e., from \(0\) to \(-2\pi\)), we can find equivalent positive angles by adding \(2\pi\) to \(t\): \[ t + 2\pi = -\frac{2\pi}{3} + \frac{6\pi}{3} = \frac{4\pi}{3} \]. This angle \(\frac{4\pi}{3}\) lies in the third quadrant.

Step 3: Find the Reference Number

In the third quadrant, the reference angle \(t_{ref}\) can be found using the formula \(\pi - t_{positive}\) when \(\pi < t_{positive} < \frac{3\pi}{2}\). Since \(t_{positive} = \frac{4\pi}{3}\), the reference number is: \[ t_{ref} = t_{positive} - \pi = \frac{4\pi}{3} - \frac{3\pi}{3} = \frac{\pi}{3}. \]

Step 4: Find the Terminal Point

The terminal point for an angle \(t\) on the unit circle is the pair \((\cos(t), \sin(t))\). To find the terminal point for \(-\frac{2\pi}{3}\), we use the cosine and sine of the reference angle \(\frac{\pi}{3}\) but adjust for the third quadrant where cosine and sine are both negative: \[ \cos \left( \frac{\pi}{3} \right) = \frac{1}{2}, \quad \sin \left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2} \] Therefore, the terminal point considering the signs in the third quadrant is: \[ \left( -\frac{1}{2}, -\frac{\sqrt{3}}{2} \right). \]

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. This makes it a valuable tool for understanding angles and their corresponding points.

• Angles are measured from the positive x-axis.
• A full rotation around the circle corresponds to an angle of \(2\pi\) radians or 360 degrees.
• The unit circle allows us to visualize angles in both positive (counter-clockwise) and negative (clockwise) directions.

With the unit circle, each point on the circle represents the terminal point of an angle \(t\). The \(x\) and \(y\) coordinates of this point represent two important trigonometric values: \(\cos(t)\) and \(\sin(t)\), respectively. This means you can easily find trigonometric values for any angle by identifying its corresponding point on the unit circle.

Terminal Point

When dealing with angles, especially on the unit circle, the terminal point is the position where the angle lands after it's been rotated from the positive x-axis.

• The terminal point is crucial for determining the sine and cosine of the angle.
• For an angle \(t\), the coordinates of the terminal point are \((\cos(t), \sin(t))\).

To illustrate, let's consider the angle \(-\frac{2\pi}{3}\). By understanding that this angle is negative, we measure it clockwise on the unit circle. Converting it to a positive equivalent, \(\frac{4\pi}{3}\), helps us locate it in the third quadrant. In this quadrant, both x and y coordinates (cosine and sine) are negative. So, the terminal point becomes \((-\frac{1}{2}, -\frac{\sqrt{3}}{2})\). This point tells us the specific trigonometric values for the given angle.

Trigonometric Values

Trigonometric values are essential components of understanding angles and their positions on the unit circle. They help describe the relationship of an angle's terminal point to the principal lines of the coordinate plane.

• \(\cos(t)\) gives us the x-coordinate, which is the horizontal distance from the origin.
• \(\sin(t)\) provides the y-coordinate, representing the vertical distance from the origin.
• These coordinates vary from -1 to 1, which aligns with the unit circle having a radius of 1.

When calculating these values for an angle like \(-\frac{2\pi}{3}\), it's the reference angle that simplifies the process. By determining the reference number \(t_{ref} = \frac{\pi}{3}\), we compute \(\cos\left(\frac{\pi}{3}\right)\) and \(\sin\left(\frac{\pi}{3}\right)\) using known trigonometric values. Adjust the signs according to the quadrant in which your angle terminates to reflect the correct position on the unit circle. For instance, in the third quadrant, both cosine and sine are negative, hence their values for \(-\frac{2\pi}{3}\) are \(-\frac{1}{2}\) and \(-\frac{\sqrt{3}}{2}\) respectively.