Give an expression that generates all angles coterminal with an angle of π/6 radian. Let n represent any integer.

Verified step by step guidance

1. Recall that angles are coterminal if they differ by an integer multiple of a full rotation. In radians, a full rotation is \(2\pi\).
2. Given the angle \(\frac{\pi}{6}\), to find all angles coterminal with it, we add multiples of \(2\pi\) to this angle.
3. Express this mathematically as \(\theta = \frac{\pi}{6} + 2\pi n\), where \(n\) is any integer (positive, negative, or zero).
4. This expression generates all angles that share the same terminal side as \(\frac{\pi}{6}\) when drawn in standard position.
5. Thus, the general formula for all coterminal angles with \(\frac{\pi}{6}\) is \(\theta = \frac{\pi}{6} + 2\pi n\), with \(n \in \mathbb{Z}\).

Key Concepts

Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides but differ by full rotations. In radians, adding or subtracting multiples of 2π to an angle results in coterminal angles. This concept helps identify all angles equivalent in position to a given angle.

Radian Measure

Radian is a unit of angular measure based on the radius of a circle. One full rotation equals 2π radians. Understanding radians is essential for expressing angles and their coterminal counterparts in a mathematically consistent way.

General Expression for Coterminal Angles

The general formula to find all angles coterminal with a given angle θ is θ + 2πn, where n is any integer. This expression accounts for all rotations around the circle, both positive and negative, generating an infinite set of coterminal angles.