Problem 44 - Algebra and Trigonometry

Question

Find (a) the reference number for each value of t, and (b) the terminal point determined by t. $$ t=\frac{13 \pi}{6} $$

Step-by-Step Solution

Answer

Reference number: \( \frac{\pi}{6} \), Terminal point: \( \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right) \).

Step 1: Understand Reference Number

In trigonometry, the reference number is the acute angle that the given angle makes with the nearest x-axis. It is within the interval \( (0, \frac{\pi}{2}) \). To find this, first determine where the angle falls on the unit circle.

Step 2: Find Equivalent Angle

Convert \( t = \frac{13\pi}{6} \) to an equivalent angle within \( [0, 2\pi) \) by subtracting \( 2\pi \), since \( \frac{13\pi}{6} > 2\pi \).\( \frac{13\pi}{6} - 2\pi = \frac{13\pi}{6} - \frac{12\pi}{6} = \frac{\pi}{6} \).

Step 3: Determine the Reference Number

The given \( t = \frac{13\pi}{6} \) converts to \( \frac{\pi}{6} \), which itself is the reference number because it is already in the first quadrant and less than \( \frac{\pi}{2} \).

Step 4: Identify Quadrant and Terminal Point

Since the angle \( \frac{\pi}{6} \) falls in the first quadrant, it does not need to be adjusted further. The terminal point for this angle corresponds to where it intersects the unit circle.

Step 5: Calculate Terminal Point Coordinates

For an angle \( \frac{\pi}{6} \), the terminal point on the unit circle is calculated using the sine and cosine functions: \( \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} \) and \( \sin(\frac{\pi}{6}) = \frac{1}{2} \). Therefore, the terminal point is \( \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right) \).

Key Concepts

Reference Number

When working with trigonometry, the concept of a reference number is crucial. A reference number is essentially an acute angle that your specific angle forms with the closest x-axis.

• This angle is always between 0 and \( \frac{\pi}{2} \) (in radians).
• It helps to simplify problems by allowing the analysis of an angle as if it were smaller and less complex.

To find the reference number for a given angle, you must first ensure that the angle is in the standard range of the unit circle, which is from 0 to \( 2\pi \). If the angle is larger than \( 2\pi \), as in the case of \( t = \frac{13\pi}{6} \), you need to convert it by subtracting multiples of \( 2\pi \) until it fits within this range. This conversion results in \( t = \frac{\pi}{6} \).

Since \( \frac{\pi}{6} \) already falls within the first quadrant, and is less than \( \frac{\pi}{2} \), it itself becomes the reference number. Understanding reference numbers can help in learning how angles behave and are analyzed within trigonometry.

Unit Circle

The unit circle is a fundamental concept in trigonometry.

• It is a circle with a radius of 1, centered at the origin (0,0) of a coordinate plane.
• It is used to define trigonometric functions for all real numbers.

The value of an angle, when positioned on the unit circle, determines a point along the circumference. Every angle has a corresponding coordinate, making the unit circle a handy tool to quickly find trigonometric function values like sine and cosine. For instance, the angle \( \frac{\pi}{6} \) determines a point on the unit circle, calculated by its cosine and sine values.

On the unit circle, important trigonometric values are derived from angles corresponding to common fractions of degrees, such as \( \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3} \), etc. A significant advantage of using the unit circle is that, due to its radius of 1, the calculations become straightforward and systematic.

Terminal Point

The terminal point is where the terminal side of an angle intersects the unit circle. To find this point,

• Determine the coordinates using trigonometric functions, specifically cosine for the x-coordinate and sine for the y-coordinate.
• For example, the terminal point of \( \frac{\pi}{6} \) on the unit circle involves the calculation: \( \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} \) and \( \sin(\frac{\pi}{6}) = \frac{1}{2} \).

These coordinates correspond to the terminal point: \( \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right) \).

This point gives us practical insight into the angle's position on the circle. It also allows us to determine the trigonometric values directly, facilitating the application of trigonometry in various problems.