Problem 46 - Algebra and Trigonometry

Question

\(39-52=\) 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

The terminal point is \(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\).

Step 1: Understand the Problem

The problem involves finding the terminal point on the unit circle for the angle \(t = \frac{13\pi}{6}\). First, we need to determine its reference angle and the corresponding terminal point.

Step 2: Find Reference Angle

The reference angle for \(t\) can be found by subtracting \(2\pi\) (which is a full rotation on the unit circle) until \(t\) is within the interval \([0, 2\pi)\). For \(t = \frac{13\pi}{6}\):\[ \frac{13\pi}{6} - 2\pi = \frac{13\pi}{6} - \frac{12\pi}{6} = \frac{\pi}{6}. \]The reference angle is \(\frac{\pi}{6}\).

Step 3: Determine Quadrant

Since \(\frac{13\pi}{6}\) reduced by \(2\pi\) becomes \(\frac{\pi}{6}\), which is less than \(\pi\) but more than \(\frac{3\pi}{2}\), the angle is in the first quadrant where both sine and cosine are positive.

Step 4: Find Terminal Point

The terminal point on the unit circle for a reference angle of \(\frac{\pi}{6}\) is known to be:\[(\cos(\frac{\pi}{6}), \sin(\frac{\pi}{6})) = \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right).\] Therefore, the terminal point for \(t = \frac{13\pi}{6}\) is \(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry used to understand angles and their corresponding trigonometric values.

• It is a circle with a radius of 1, centered at the origin of a coordinate plane.
• The unit circle allows us to easily determine sine and cosine values for different angles.
• Angles on the unit circle are typically measured in radians.
  • One complete revolution around the circle corresponds to an angle of \(2\pi\) radians.

The key to using the unit circle effectively is to understand how angles correspond to points on the circle. Each position on the unit circle represents an angle, denoted by \(t\), from the initial side (positive x-axis) to the terminal side (where the angle ends).Understanding this helps us find the sine (y-coordinate) and cosine (x-coordinate) values at that angle, which are crucial for trigonometric calculations. For example, for the angle \(\frac{\pi}{6}\), the corresponding point is \((\frac{\sqrt{3}}{2}, \frac{1}{2})\), which reflects the values of \(\cos(\frac{\pi}{6})\) and \(\sin(\frac{\pi}{6})\).

Trigonometric Quadrants

Trigonometric quadrants are critical for determining the sign of trigonometric functions like sine and cosine based on the angle's location. The coordinate plane is divided into four quadrants by the x-axis and y-axis.

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive, cosine is negative.
• Quadrant III: Both sine and cosine are negative.
• Quadrant IV: Sine is negative, cosine is positive.

Identifying the quadrant of an angle helps determine the sign of its sine and cosine. For an angle like \(\frac{\pi}{6}\), which is within the first quadrant, both sine and cosine are positive. This explains why the terminal point for the angle \(t = \frac{13\pi}{6}\) is \((\frac{\sqrt{3}}{2}, \frac{1}{2})\), because its reference angle falls in the first quadrant, keeping both values positive.

Terminal Point

The terminal point is the end point of an angle when drawn in standard position on the unit circle. It represents where the angle "terminates," or ends, after being drawn from the positive x-axis.

• To find the terminal point, you need to calculate the reference angle for \(t\) and determine its location on the unit circle.
• The reference angle helps pinpoint the position by reducing the initial angle to one within the first revolution \([0, 2\pi)\).

For the specific angle \(t = \frac{13\pi}{6}\), we derived the reference angle as \(\frac{\pi}{6}\) by subtracting \(2\pi\) from \(t\). Since \(\frac{\pi}{6}\) is in the first quadrant, we find that this corresponds to the point \((\frac{\sqrt{3}}{2}, \frac{1}{2})\). Thus, this point is the terminal point for the angle \(t\) on the unit circle, which connects back to understanding trigonometric values based on the quadrant of the reference angle.