Problem 49 - Precalculus Mathematics for Calculus

Question

Find (a) the reference number for each value of \(t\) and (b) the terminal point determined by \(t\). $$t=-\frac{11 \pi}{3}$$

Step-by-Step Solution

Answer

Reference number for \( t = -\frac{11\pi}{3} \) is \( \frac{\pi}{3} \); terminal point is \( \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) \).

Step 1: Understanding the Reference Number

To find the reference number for a given value of \( t \), we need to express \( t \) in terms of an angle between 0 and \( 2\pi \). The reference number is the smallest positive co-terminal angle to \( t \).

Step 2: Finding a Positive Co-terminal Angle

The given value of \( t \) is \( -\frac{11\pi}{3} \). First, find a positive co-terminal angle by adding \( 2\pi \) repeatedly until you get a positive angle. \( 2\pi = \frac{6\pi}{3} \). So we calculate:\[-\frac{11\pi}{3} + \left(2 \times \frac{6\pi}{3}\right) = -\frac{11\pi}{3} + \frac{12\pi}{3} = \frac{\pi}{3}\]Now, \( \frac{\pi}{3} \) is positive and less than \( 2\pi \). Thus, \( \frac{\pi}{3} \) is the reference number.

Step 3: Identifying the Terminal Point

The terminal point of an angle \( t \) on the unit circle is found using its reference angle (as they share the same terminal appearance). Since the reference angle \( \frac{\pi}{3} \) corresponds to the first quadrant, the coordinates (cosine and sine) at \( \frac{\pi}{3} \) are \( \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) \). Therefore, the terminal point determined by \( t = -\frac{11\pi}{3} \) is \( \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) \).

Key Concepts

Reference Number

Trigonometric angles often involve finding the reference number to simplify calculations. Think of the reference number as a handy helper to reduce any angle. It's the smallest positive co-terminal angle. This means it shares the same terminal side with your given angle, but is confined within a single rotation, between 0 and \( 2\pi \).
For example, given \( t = -\frac{11\pi}{3} \), we'll find the reference number by adding \( 2\pi \) (or \( \frac{6\pi}{3} \)) until we land on a positive angle. Here's why:

• Negative angles are rotated clockwise, positive angles counter-clockwise.
• Co-terminal angles are reached by making full rotations (in units of \( 2\pi \)).

So, repeatedly add \( 2\pi \) until the angle is positive: \[-\frac{11\pi}{3} + \frac{12\pi}{3} = \frac{\pi}{3}\] Now \( \frac{\pi}{3} \) is a positive angle, and it fits perfectly within a single complete turn (0 to \( 2\pi \)). This means it truly represents the reference number for \( t = -\frac{11\pi}{3} \).

Terminal Point

The terminal point is pivotal in understanding where an angle finishes its journey on the unit circle. It pinpoints the precise spot along the circle's edge linked to the angle.
So, when \( t = -\frac{11\pi}{3} \), find its terminal point based on its reference angle, which is \( \frac{\pi}{3} \). This reference angle lands in the first quadrant, where both cosine and sine are positive.

• The unit circle lets us access trigonometric values directly.
• The coordinates of these values act as the terminal points.

For \( \frac{\pi}{3} \)
The cosine (x-value) is \( \frac{1}{2} \)
The sine (y-value) is \( \frac{\sqrt{3}}{2} \)
Thus, the terminal point for \( t = -\frac{11\pi}{3} \) is: \( \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right)\).

Unit Circle

A cornerstone in trigonometry is the unit circle, a circle with a radius of 1 centered at the origin of the coordinate system. It's a tool that ties angles to coordinates effortlessly.
Upon the unit circle:

• The x-coordinate of a point equals the cosine of its angle.
• The y-coordinate equals the sine of the angle.
• Every point corresponds to one specific angle form, among infinite co-terminal editions.

For instance, \( \frac{\pi}{3} \) sits in the first quadrant, where:- Its cosine is \( \frac{1}{2} \).- Its sine is \( \frac{\sqrt{3}}{2} \).
This makes the coordinates \( \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) \) easy to retrieve by simply knowing the angle. The unit circle is a roadmap for angles, revealing where they begin, complete a rotation, and ends with terminal points—ideal for solving trigonometric problems with precision.