Problem 41 - Precalculus Mathematics for Calculus

Question

Find (a) the reference number for each value of \(t\) and (b) the terminal point determined by \(t\). $$t=\frac{3 \pi}{4}$$

Step-by-Step Solution

Answer

Reference angle: \(\frac{\pi}{4}\), Terminal point: \((-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\).

Step 1: Understanding the Problem

We need to find the reference number and the terminal point for the given angle \(t = \frac{3\pi}{4}\). This involves determining the reference angle in the unit circle and finding the coordinates on the unit circle that correspond to this angle.

Step 2: Determine the Quadrant

The angle \(t = \frac{3\pi}{4}\) is located in the second quadrant. This is because \(\frac{3\pi}{4}\) is greater than \(\frac{\pi}{2}\) (\(90^\circ\)) and less than \(\pi\) (\(180^\circ\)).

Step 3: Find the Reference Angle

The reference angle for an angle in the second quadrant is calculated using the formula: \( \pi - t \). Plugging in the value, we get \( \pi - \frac{3\pi}{4} = \frac{\pi}{4} \). Thus, the reference angle is \(\frac{\pi}{4}\).

Step 4: Determine the Terminal Point

To find the terminal point, identify the coordinates on the unit circle that correspond to \(t = \frac{3\pi}{4}\). In the second quadrant, the coordinates will be of the form \((-x, y)\). Since the reference angle is \(\frac{\pi}{4}\), the coordinates for the terminal point are \(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry and a great tool for understanding angles and their corresponding coordinates. Think of it as a circle with a radius of 1 unit, centered at the origin (0,0) on a coordinate plane. This simple structure makes it easier to work with trigonometric functions.
The unit circle helps us visualize angles and terminal points, which are essential for understanding the behavior of sine and cosine values.

• Each point on the unit circle is associated with an angle, measured from the positive x-axis.
• The x-coordinate of a point is the cosine of the angle, while the y-coordinate is the sine.

For example, when you have an angle like \(t = \frac{3\pi}{4}\), it lies on the unit circle and helps us find its corresponding sine and cosine values. As the radius is 1, trigonometric identities, such as \( \sin^2 \theta + \cos^2 \theta = 1 \), are apparent just by observing the unit circle. This visualization aids in understanding and calculating different trigonometric functions effectively.

Reference Angle

The reference angle is an indispensable tool in trigonometry that allows us to compute trigonometric functions more easily by referencing a known acute angle. Essentially, it is the angle formed by dropping a perpendicular from the terminal side of the angle to the x-axis.
Reference angles are always positive and lie between 0 and \(\frac{\pi}{2}\) radians (0 to \(90^\circ\)).

• In the context of the unit circle, the reference angle allows for quick determination of the trigonometric values in other quadrants.
• To find the reference angle for any angle \(t\), you subtract \(t\) from \(\pi\) in the second quadrant or adjust accordingly for other quadrants.
• For our example where \(t = \frac{3\pi}{4}\), the reference angle is \(\pi - \frac{3\pi}{4} = \frac{\pi}{4}\).

Understanding the reference angle is essential as it simplifies the process of finding sine and cosine values for angles in different quadrants. It transforms complex angle calculations into simpler reference-based computations.

Terminal Point

The terminal point is where an angle \(t\) intersects the unit circle, and it corresponds to specific sine and cosine values.
This is crucial for understanding trigonometric values since these points define where the angle `lands` on the circle.

• When you know an angle \(t\), you can determine its terminal point using the unit circle.
• By using trigonometric functions derived from the reference angle, you can identify the coordinates of the terminal point.
• In the second quadrant, for instance, the x-coordinate becomes negative, affecting the terminal point's location.
• For \(t = \frac{3\pi}{4}\), the terminal point is \((-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\), as the angle is in the second quadrant.

Comprehending the terminal point on the unit circle helps us predict and calculate the corresponding trigonometric function values, aiding in graphing and analyzing periodic functions.