Problem 30 - Algebra and Trigonometry

Question

The measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle. $$ \frac{11 \pi}{6} $$

Step-by-Step Solution

Answer

Positive: \(\frac{23\pi}{6}, \frac{35\pi}{6}\); Negative: \(-\frac{\pi}{6}, -\frac{13\pi}{6}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position on the coordinate plane. To find coterminal angles, we can add or subtract full rotations. In radians, a full rotation is represented by \(2\pi\).

Step 2: Calculating Positive Coterminal Angles

To find positive coterminal angles with the given angle \(\frac{11\pi}{6}\), we add full rotations. Since a rotation is \(2\pi\), we calculate the positive angles as follows: \(\frac{11\pi}{6} + 2\pi = \frac{11\pi}{6} + \frac{12\pi}{6} = \frac{23\pi}{6}\). For the second positive coterminal angle, add another \(2\pi\): \(\frac{23\pi}{6} + \frac{12\pi}{6} = \frac{35\pi}{6}\).

Step 3: Calculating Negative Coterminal Angles

To find negative coterminal angles with \(\frac{11\pi}{6}\), we subtract full rotations: \(\frac{11\pi}{6} - 2\pi = \frac{11\pi}{6} - \frac{12\pi}{6} = -\frac{\pi}{6}\). For the second negative coterminal angle, subtract another \(2\pi\): \( -\frac{\pi}{6} - \frac{12\pi}{6} = -\frac{13\pi}{6}\).

Key Concepts

Angle Measurement

Angle measurement is a fundamental concept in geometry and trigonometry. Angles are a way to describe the rotation needed to direct from a given position to another. They can be measured in several units, with degrees and radians being the most common.

An angle is formed between two rays that originate from a common endpoint, known as the vertex. Different systems of measurement describe angles:

• Degrees: A full circle is divided into 360 degrees.
• Radians: The angle formed by taking the radius of a circle and wrapping it along the circle's edge. A full circle is equivalent to \(2\pi\) radians.

Both measurements serve specific purposes and allow for flexibility in descriptions of rotations and calculations. Understanding angle measurements is crucial for solving problems related to coterminal angles, as it involves manipulation of these values to find equivalent rotations.

Radian Measure

Radian measure offers a unique and efficient way to describe angles, especially in mathematics and physics. Since a radian relates directly to the radius of a circle, it naturally connects angle measures to the geometry of circles.

One radian is defined by the angle subtended by an arc that is equal in length to the radius of the circle. To understand this concept further:

• There are \(2\pi\) radians in a complete circle.
• \(\pi\) radians corresponds to 180 degrees.

For example, an angle given as \(\frac{11\pi}{6}\) radians represents a significant portion of a full circle (almost a full rotation). It is especially useful in calculus and trigonometry because of the seamless link between linear and circular dimensions it provides. Radians simplify many mathematical expressions and help to find coterminal angles effectively by using increments of \(2\pi\).

Angle Rotation

Angle rotation is integral to understanding coterminal angles. When you create coterminal angles, you essentially rotate one angle by a full circle (or multiples of it).

The concept is defined by adding or subtracting full rotations to a given angle:

• A full rotation in radians is \(2\pi\).
• Positive rotations occur in the counterclockwise direction.
• Negative rotations move clockwise.

Coterminal angles share the same terminal side after one or more complete rotations. For example, starting with an angle measure of \(\frac{11\pi}{6}\), adding \(2\pi\) results in \(\frac{23\pi}{6}\), a positive coterminal angle. Similarly, subtracting \(2\pi\) gives \(-\frac{\pi}{6}\), a negative coterminal angle. This method allows the discovery of multiple angle measures that have the same spatial orientation.