Problem 48 - Algebra and Trigonometry

Question

Find (a) the reference number for each value of t, and (b) the terminal point determined by t. $$ t=\frac{31 \pi}{6} $$

Step-by-Step Solution

Answer

(a) Reference number: \( \frac{7\pi}{6} \). (b) Terminal point: \((-\frac{\sqrt{3}}{2}, -\frac{1}{2})\).

Step 1: Understanding the Problem

We are given the angle \( t = \frac{31\pi}{6} \) and asked to find both the reference number and the terminal point on the unit circle. The reference number is the smallest positive coterminal angle between \( 0 \) and \( 2\pi \), and the terminal point is the location on the unit circle corresponding to this angle.

Step 2: Finding the Reference Number

The reference number is the smallest positive coterminal angle within \([0, 2\pi)\). To find it, we reduce \( \frac{31\pi}{6} \) by subtracting \( 2\pi \) multiple times until it is within this range.

Step 3: Calculating Coterminal Angles

Subtract \( 2\pi \) from \( \frac{31\pi}{6} \):\[\frac{31\pi}{6} - 2\pi = \frac{31\pi}{6} - \frac{12\pi}{6} = \frac{19\pi}{6}\]\(\frac{19\pi}{6} > 2\pi \), repeat:\[\frac{19\pi}{6} - 2\pi = \frac{19\pi}{6} - \frac{12\pi}{6} = \frac{7\pi}{6}\]Now \( \frac{7\pi}{6} \) is within \([0, 2\pi)\), so it is the reference number.

Step 4: Finding the Terminal Point

The reference number \( \frac{7\pi}{6} \) is in the third quadrant, where sine and cosine are both negative. The angle is \( \pi + \frac{\pi}{6} \), so the terminal point is:\[\left( -\cos\left(\frac{\pi}{6}\right), -\sin\left(\frac{\pi}{6}\right) \right) = \left( -\frac{\sqrt{3}}{2}, -\frac{1}{2} \right)\]

Step 5: Final Result

The reference number for \( t = \frac{31\pi}{6} \) is \( \frac{7\pi}{6} \), and the terminal point is \( \left( -\frac{\sqrt{3}}{2}, -\frac{1}{2} \right) \).

Key Concepts

Reference Angle

A reference angle is a concept that simplifies the analysis of angles by relating them to angles within the first quadrant. For any angle \(\theta\), the reference angle is always a positive acute angle, i.e., between \(0\) and \(\pi/2\) radians (or \(0\) and \(90\) degrees). It is used to determine the trigonometric values of the original angle by reflecting it within the first quadrant.

• To find the reference angle for an angle in standard position, identify the smallest angle between the terminal side of the angle and the x-axis.
• Reference angles are particularly useful because they help in visualizing and calculating trigonometric functions in all four quadrants.
• In the context of \(\frac{31\pi}{6}\), we find the reference angle by reducing it to within the \(0\) to \(2\pi\) range, leading us to \(\frac{7\pi}{6}\) as a significant step.

Unit Circle

The unit circle is a vital concept in trigonometry that represents all the possible angles and their trigonometric functions on a circle with a radius of one. The circle is essential because it simplifies the calculation of trigonometric functions for any angle by mapping them onto coordinates on this circle.

• Each point on the unit circle corresponds to an angle whose vertex is at the circle's center and sides intersect the circle's circumference.
• The x-coordinate of a point is the cosine of the angle, while the y-coordinate is its sine.
• Mentioning the unit circle in solving for \(t = \frac{31\pi}{6}\), it allowed us to find the terminal point, which gives the positional coordinates for the angle on this circle.

Understanding the unit circle helps in grasping how angles rotate and how their sine and cosine values fluctuate throughout the circle, which is critical when comparing or calculating coterminal and reference angles.

Terminal Point

In trigonometry, the terminal point is the final location of an angle on the unit circle once it completes its rotation from the positive x-axis.

• The terminal point is where the angle stops traveling on the circumference of the unit circle.
• The coordinates of the terminal point provide the sine and cosine values for that angle.
• In our example with \(t = \frac{31\pi}{6}\), the terminal point was calculated to be \(\left( -\frac{\sqrt{3}}{2}, -\frac{1}{2} \right)\). These coordinates tell us the angle's sine and cosine values, both negative, which places the terminal point in the third quadrant of the unit circle.

Knowing the terminal point for an angle is invaluable as it specifies the exact coordinate that corresponds to the angle on the unit circle, encompassing both its horizontal and vertical components.

Coterminal Angles

Coterminal angles are angles that share the same terminal side but may have different measures. This happens when you add or subtract full \(2\pi\) radians (or \(360\) degrees) from an angle.

• Coterminal angles are helpful when working to find an angle within a standard range, usually between \(0\) and \(2\pi\) radians.
• They allow us to express angles in different forms while maintaining their foundational trigonometric properties.
• With \(t = \frac{31\pi}{6}\), by subtracting \(2\pi\) until the angle was in \([0, 2\pi)\), we found the coterminal angle \(\frac{7\pi}{6}\) which directly guided us to identify the terminal point.

Understanding coterminal angles is crucial because they demonstrate how periodic trigonometric functions are and allow transformations of complex angles into simpler, more manageable terms.