Problem 22 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Sketch the angle. Then find its reference angle. \(-\frac{13 \pi}{6}\)

Step-by-Step Solution

Answer

The angle \(-\frac{13 \pi}{6}\) is equivalent to \(-\frac{\pi}{6}\) in the range from 0 to \(2\pi\), and its sketch will be in the 4th quadrant, just below the x-axis. Its reference angle is \(\frac{\pi}{6}\).

Step 1: Convert the Negative Angle to Positive

Given the angle \(-\frac{13 \pi}{6}\), let's find its positive equivalent. We can get this by adding \(2\pi\) (in terms of sixths, that's adding \(\frac{12\pi}{6}\)). So we do this calculation: \[-\frac{13 \pi}{6} + \frac{12\pi}{6} = -\frac{\pi}{6}\]This means an equivalent positive rotation is \(-\frac{\pi}{6}\) from \(2\pi\).

Step 2: Sketch the Angle

On a clockwise rotation on the unit circle from the positive x-axis, \(-\frac{\pi}{6}\) would be an angle in the 4th quadrant. The rotation would be \(\frac{\pi}{6}\) or 30 degrees just below the x-axis.

Step 3: Determine the Reference Angle

The reference angle is the acute (positive and less than 90 degrees) angle that's formed between the terminal side of an angle in standard position and the x-axis. For an angle in the 4th quadrant, like \(-\frac{\pi}{6}\), the reference angle is its positive equivalent. So, the reference angle here is \(\frac{\pi}{6}\).

Key Concepts

Reference Angle Theory

In trigonometry, the reference angle is a crucial concept that helps simplify finding the sine, cosine, and tangent of any angle. The reference angle is the smallest angle that can project the terminal side of a given angle to the x-axis.

• The reference angle always has a value between 0 and \(\frac{\pi}{2}\) radians (or 0 to 90 degrees).
• It is always positive and measured in the counterclockwise direction from the x-axis.
• Importantly, the reference angle is made with the nearest x-axis, not necessarily the horizontal axis.

For example, if you're dealing with an angle in the fourth quadrant, like our original question of \(-\frac{13\pi}{6}\), the reference angle is found by calculating the smallest angle between the terminal side and the positive x-axis, resulting in \(\frac{\pi}{6}\). This process ensures that all associated calculations using the trigonometric functions are manageable and streamlined.

Understanding the Unit Circle

The unit circle is a key concept to master in trigonometry, especially when working with angles measured in radians. The circle has a radius of exactly one unit centered at the origin of a coordinate plane, and it is divided into four quadrants.

• All positions on the unit circle can be represented as points (x, y), where x is the cosine of the angle and y is the sine of the angle.
• The circumference of the unit circle is \(2\pi\) or 360 degrees because this is a full rotation around the circle.
• Each quadrant of the circle corresponds to a quarter of these values \((\frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi)\).

This structure allows any angle to be plotted by its rotation from the positive x-axis. For example, with an angle such as \(-\frac{13\pi}{6}\), it is plotted by determining its equivalent positive rotation \(-\frac{\pi}{6}\) in the coordinate system. This will help in visualizing how angles correspond to positions on the unit circle.

Navigating the Quadrants

The Cartesian plane is divided into four sections called quadrants, each defined by the x and y axes intersecting at the origin. Understanding these quadrants helps in determining reference angles and interpreting the sign of trigonometric functions.

• Quadrant I: Both x and y coordinates are positive. Angle measures range from 0 to \(\frac{\pi}{2}\).
• Quadrant II: The x coordinate is negative and the y coordinate is positive. Angle measures range from \(\frac{\pi}{2}\) to \(\pi\).
• Quadrant III: Both coordinates are negative. Angle measures range from \(\pi\) to \(\frac{3\pi}{2}\).
• Quadrant IV: The x coordinate is positive, and the y is negative. Angle measures range from \(\frac{3\pi}{2}\) to \(2\pi\).

For an angle like \(-\frac{\pi}{6}\), found in Quadrant IV, recognizing its position is essential. This helps not only with sketching the angle accurately but also in understanding how it resolves to a reference angle \(\frac{\pi}{6}\), and how sine and cosine share different signs in this quadrant. The more you practice, the more intuitive these quadrants and their characteristics will become.