Problem 46 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

Reference number: \(\frac{1\pi}{6}\). Terminal point: \((\frac{ rac{ }{\underline{\phantom{xx}}} rac{ }{\underline{\phantom{xx}}} rac{ }{\underline{\phantom{xx}}}3}{2}, \frac{1}{2})\)."}

Step 1: Recognize Period of the Circle

The circle completes one rotation at \(2\pi\). For trigonometric problems involving angles, consider the angle modulo \(2\pi\) to determine the equivalent angle within one rotation of the circle.

Step 2: Calculate Modulo to Find Reference Number

Calculate \(\frac{13\pi}{6} \mod 2\pi\). To find \(2\pi\) in terms of sixths, we multiply to find \(2\pi = \frac{12\pi}{6}\). We then subtract \(\frac{12\pi}{6}\) from \(\frac{13\pi}{6}\) to get \(\frac{1\pi}{6}\). Thus, the reference number is \(\frac{rac{}{\underline{\phantom{xx}}}1 \pi}{6}\), which is equivalent to \(\frac{rac{}{\underline{\phantom{xx}}}1\pi}{6}\).

Step 3: Determine Terminal Point Based on Reference Number

Since the reference number \(\frac{rac{}{\underline{\phantom{xx}}}1 \pi}{6}\) represents an angle in the terminal side quadrant, we find the point on the unit circle determined by this angle. The angle \(\frac{rac{}{\underline{\phantom{xx}}}1\pi}{6}\) corresponds to 30 degrees. Thus, the terminal point is \((\cos(\frac{rac{}{\underline{\phantom{xx}}}1\pi}{6}), \sin(\frac{rac{}{\underline{\phantom{xx}}}1\pi}{6}))\).

Step 4: Compute Values of Trigonometric Functions

From trigonometric tables, \(\cos(\frac{rac{}{\underline{\phantom{xx}}}1\pi}{6}) = \frac{rac{}{\underline{\phantom{xx}}}rac{}{\underline{\phantom{xx}}}rac{}{\underline{\phantom{xx}}}rac{}{\underline{\phantom{xx}}}3}{2\) and \(\sin(\frac{rac{}{\underline{\phantom{xx}}}1\pi}{6}) = \frac{rac{}{\underline{\phantom{xx}}}1}{rac{}{\underline{\phantom{xx}}}rac{}{\underline{\phantom{xx}}}rac{}{\underline{\phantom{xx}}}2}\). Therefore, the terminal point is \((\frac{rac{}{\underline{\phantom{xx}}}rac{}{\underline{\phantom{xx}}}rac{}{\underline{\phantom{xx}}}3}{rac{}{\underline{\phantom{xx}}}rac{}{\underline{\phantom{xx}}}rac{}{\underline{\phantom{xx}}}2}, \frac{rac{}{\underline{\phantom{xx}}}1}{rac{}{\underline{\phantom{xx}}}rac{}{\underline{\phantom{xx}}}rac{}{\underline{\phantom{xx}}}2}\).

Key Concepts

Reference Angle

A reference angle helps us identify the position of an angle on the unit circle, making it easier to determine trigonometric functions. It is always the smallest angle between the terminal side of the angle and the horizontal axis. Reference angles simplify calculations by keeping the focus on the smaller, more manageable portion of an angle within a circle.

When you have an angle, such as \(\frac{13\pi}{6}\), that is larger than \(2\pi\) (a full rotation), you first bring it back within the first circle rotation using the concept of modulo, which we’ll cover later. The reference angle is the acute angle you obtain here.

• A reference angle is always between \(0\) and \(\frac{\pi}{2}\) radians.
• In this problem, the reference angle \(\frac{\pi}{6}\) helps us find values on the unit circle by correlating angles to their more familiar values within one circle rotation.

Unit Circle

The unit circle is crucial when working with trigonometric functions because it helps visualize angles and their trigonometric values. Defined as a circle with a radius of one, the unit circle lays the foundation for trigonometric function definitions and their properties.

Each point on the unit circle corresponds to the angle's cosine and sine values from the center of the circle. For instance, in our example with \(\frac{\pi}{6}\):

• Cosine value \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\)
• Sine value \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\)

These intersection points on the circle help provide exact trigonometric ratios, such as those seen when reaching \(\frac{13\pi}{6}\), and eventually transform into reference angles that provide terminal points.

Angle Modulo

The concept of using modulo in trigonometry mainly revolves around simplifying angles to maintain them within a primary circle's rotation. Since a full circle is \(2\pi\) radians, any angle that is larger than \(2\pi\) can be redefined into an equivalent position using modulo. This method helps locate the angle within one rotation of the unit circle, which in our case is key for proper reference angle calculation.

For example, if you take an angle like \(\frac{13\pi}{6}\), you follow these steps:

• First, express \(2\pi\) in a common fraction form, \(\frac{12\pi}{6}\).
• Then, subtract this from your given angle: \(\frac{13\pi}{6} - \frac{12\pi}{6} = \frac{\pi}{6}\).

The result \(\frac{\pi}{6}\) is the angle modulo \(2\pi\). It represents the same position on the unit circle as \(\frac{13\pi}{6}\), thus aiding in finding the reference angle and ultimately the correct terminal point on the unit circle.