Problem 41 - Algebra and Trigonometry

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: Convert to Positive Angle

Since the angle given is negative, we first need to convert it to a positive angle within the standard circle. A full circle is represented by an angle of \(2\pi\). We can find the equivalent positive angle by adding \(2\pi\) to the given angle: \[t' = -\frac{2\pi}{3} + 2\pi = \frac{4\pi}{3}\]This positive angle will now be used to determine the reference number and terminal point.

Step 2: Find the Reference Number

The reference number is the acute angle between the terminal side of the angle and the x-axis. Since \(t' = \frac{4\pi}{3}\) lies in the third quadrant, the reference number is calculated as:\[\text{Reference Number} = \pi - \left(\frac{4\pi}{3} - \pi\right) = \frac{\pi}{3}\]This is because the angle \(\frac{4\pi}{3}\) can be seen as \(\pi + \frac{\pi}{3}\).

Step 3: Find Terminal Point

The terminal point of an angle \(t\) is the point on the unit circle determined by traveling a distance \(t\) counterclockwise from the point (1,0). For \(t = \frac{4\pi}{3}\), use the sine and cosine values for the reference angle:- Since \(\frac{4\pi}{3}\) is in the third quadrant, both sine and cosine are negative.- Thus, \(\cos\left(\frac{4\pi}{3}\right) = -\cos\left(\frac{\pi}{3}\right) = -\frac{1}{2}\)- And \(\sin\left(\frac{4\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}\)So, the terminal point is \(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\).

Key Concepts

Reference Angle

Understanding reference angles is crucial in trigonometry. A reference angle is always positive and less than or equal to \(\frac{\pi}{2}\). It gives us a way to find equivalent sine and cosine values for different angles, while always reverting to a first-quadrant angle. But how do you establish a reference angle for an unfamiliar angle like \(t = -\frac{2 \pi}{3}\)?
Start by identifying the positive equivalent of the angle. This transformation helps fit the angle inside the standard unit circle range of 0 to \(2\pi\). In this case, we found our equivalent angle to be \(t' = \frac{4\pi}{3}\). The next step is key: realize where the angle digitized to the third quadrant in the circle. Here, the reference angle is determined by the separation between this angle and the x-axis. By viewing \(\frac{4\pi}{3}\) as \(\pi + \frac{\pi}{3}\), we identify that the reference angle is a simple, more recognizable value: \(\frac{\pi}{3}\).
Aiding with complex problem solving, always keep reference angles straightforward and note:

• They help simplify trigonometric expression evaluations.
• A reference angle cannot exceed \(90^\circ\) or \(\frac{\pi}{2}\).
• They support identifying sine, cosine values.

Unit Circle

The unit circle serves as a fundamental framework in trigonometry. It aids in understanding how angles relate to coordinate systems and defines the trigonometric functions: sine and cosine. A unit circle, by definition, is a circle of radius 1 centered at the origin \((0,0)\) in the Cartesian plane. As you travel around the unit circle, each angle \(t\) from the positive x-axis correlates with a coordinate point \((\cos(t), \sin(t))\).
When solving the exercise problem with \(t = \frac{4\pi}{3}\), we are discovering this particular coordinate point on the unit circle. Knowing which quadrant the angle is in ensures accuracy for the signs of \(\cos(t)\) and \(\sin(t)\).
Here's the leverage of the unit circle:

• It visualizes outcomes of trigonometric functions intuitively.
• All angle-based transformations are seamlessly represented.
• Supplementary relations like \(\tan(t) = \frac{\sin(t)}{\cos(t)}\) can be derived.

Revisiting this concept aids in recognizing patterns universal across the entire circle. This insight streamlines understanding and application, especially when angles enter different quadrants.

Terminal Point

To wrap up the concept journey, let's examine terminal points. The terminal point provides the final position on the unit circle after moving through an angle \(t\). The logic is straightforward: imagine starting at \((1, 0)\) and incrementally moving up counterclockwise along the circle.
In context, for \( \frac{4\pi}{3} \), start from \((1,0)\) and move with the angle's arc to fall in the third quadrant. Considering this direction reveals crucial properties of trigonometric signs—negative in both sine and cosine for the third quadrant.
With our solved example:
The trigonometric calculation helped us narrow down the terminal point to \((-\frac{1}{2}, -\frac{\sqrt{3}}{2})\). This point results from the earlier deduced values of \(-\cos(\frac{\pi}{3})\) and \(-\sin(\frac{\pi}{3})\). Important factors when noting the terminal point are:

• It essentially defines the angle's unique place on the circle.
• The coordinate highlights the unit circle's fundamental properties.
• It showcases quadrant sign rules.

Employing terminal points aids in consolidating the unit circle comprehension and solidifying the angle's correlation with trigonometric results.