Problem 62 - A Graphical Approach to Precalculus with Limits

Question

Give an expression that generates all angles coterminal with each angle. Let \(n\) represent any integer. $$\frac{\pi}{6}$$

Step-by-Step Solution

Answer

The expression is \(\theta = \frac{\pi}{6} + 2n\pi\), where \(n\) is any integer.

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same terminal side. This occurs when you add or subtract full rotations (i.e., multiples of a circle) to/from an angle. In the context of radians, a full rotation is \(2\pi\).

Step 2: Identify the Initial Angle

The given angle is \(\frac{\pi}{6}\). This is our initial angle from which we'll derive coterminal angles.

Step 3: Determine the General Expression for Coterminal Angles

To generate all angles coterminal with \(\frac{\pi}{6}\), we add or subtract full rotations (\(2\pi\)) to the given angle. Therefore, the general expression is \( \theta = \frac{\pi}{6} + 2n\pi \), where \(n\) is any integer.

Step 4: Finalize the Expression

With \(n\) as any integer, the expression \( \theta = \frac{\pi}{6} + 2n\pi \) captures all possible angles that are coterminal with \(\frac{\pi}{6}\). This expression accounts for both positive and negative rotations.

Key Concepts

Radians

When working with angles in mathematics, especially in trigonometry, radians provide a natural and intuitive way to measure them. Unlike degrees, which divides the circle into 360 equal parts, radians measure angles based on the radius of the circle. One full circle in radians is equivalent to \(2\pi\).

• Radians simplify many equations, especially those involving circles and periodic functions.
• They allow us to express angle rotations in a format that's more related to the geometry of circles.

For example, \(\frac{\pi}{6}\) radians can be visualized as a slice of a circle, akin to cutting a pizza into 12 equal parts and taking one slice. This perspective makes radians immensely useful when we perform calculations involving circumference and arc length, because it directly relates the angle to the radius.

Angle Rotation

The concept of angle rotation is fundamental in understanding how coterminal angles work. Rotations can be performed either in a clockwise or counterclockwise direction. When you rotate a ray from the x-axis, the end point travels along a circle, forming angles with different rays from the center point.

• Clockwise rotations result in negative angles.
• Counterclockwise rotations result in positive angles.

By recognizing these rotations and the directionality, we can derive angles that share the same terminal side, known as coterminal angles. When we add \(2\pi\) to an angle, we complete one full circle, returning to a position that is coterminal, or located at the same ending point on the circle. Understanding the movement along this circular path is crucial for calculating coterminal angles successfully.

Full Rotation

A full rotation signifies a complete circular movement that brings a point back to its original position. In the context of radians, a full rotation is represented by \(2\pi\).

• Completing a rotation of \(2\pi\) brings an angle back to its starting position.
• This allows us to generate coterminal angles by adding or subtracting multiples of \(2\pi\).

For example, if you start at \(\frac{\pi}{6}\) and add \(2\pi\), you arrive at a coterminal angle. Similarly, subtracting \(2\pi\) brings you back to a coterminal angle but in a negative direction. The beauty of understanding full rotations is in its ability to visually and mathematically link different expressions of the same angle, making it easier to manipulate and understand periodicity in functions.