Angles, Coterminal Angles, and Rotations in Precalculus

Angles and Coterminal Angles

Understanding Angles and Rotations

In precalculus, angles are measured in radians or degrees and represent the amount of rotation from an initial side to a terminal side. Coterminal angles are angles that share the same initial and terminal sides but may differ by full rotations (multiples of $2\pi$ radians or $360^\circ$).

• Angle: The measure of rotation between two rays sharing a common endpoint (vertex).

• Coterminal Angles: Angles that have the same terminal side when drawn in standard position.

• Full Rotation: $2\pi$ radians or $360^\circ$.

Finding Coterminal Angles

To find a coterminal angle, add or subtract multiples of $2\pi$ radians (or $360^\circ$) to the given angle.

• Formula: $\theta_{coterminal} = \theta + 2\pi k$, where $k$ is any integer.

• Example: Given $\frac{23\pi}{5}$, subtract $2\pi$ to find a coterminal angle within one full rotation:

$\frac{23\pi}{5} - 2\pi = \frac{23\pi}{5} - \frac{10\pi}{5} = \frac{13\pi}{5}$

• $\frac{13\pi}{5}$ is still greater than $2\pi$, so subtract another $2\pi$:

$\frac{13\pi}{5} - 2\pi = \frac{13\pi}{5} - \frac{10\pi}{5} = \frac{3\pi}{5}$

• Conclusion: $\frac{23\pi}{5}$ is coterminal with $\frac{3\pi}{5}$.

Negative Angles and Coterminal Angles

Negative angles represent clockwise rotation. To find a positive coterminal angle, add $2\pi$ until the result is positive.

• Example: Given $-\frac{15\pi}{5}$:

$-\frac{15\pi}{5} + 2\pi = -\frac{15\pi}{5} + \frac{10\pi}{5} = -\frac{5\pi}{5} = -\pi$

• Still negative, add another $2\pi$:

$-\pi + 2\pi = \pi$

• Conclusion: $-\frac{15\pi}{5}$ is coterminal with $\pi$.

Visual Representation of Coterminal Angles

On the unit circle, coterminal angles share the same terminal side. The diagrams show multiple rotations, with the original angle and its coterminal angle ending at the same position.

• Key Point: Angles differing by full rotations ($2\pi$ radians) are coterminal.

• Application: Useful for simplifying trigonometric expressions and solving equations.

Summary Table: Coterminal Angle Calculations

Original Angle	Operation	Coterminal Angle
$\frac{23\pi}{5}$	Subtract $2\pi$ twice	$\frac{3\pi}{5}$
$-\frac{15\pi}{5}$	Add $2\pi$ twice	$\pi$

Additional info:

• Angles greater than $2\pi$ or less than $0$can always be reduced to a coterminal angle between $0$ and $2\pi$ by adding or subtracting $2\pi$ as needed.

• This process is essential for understanding periodicity in trigonometric functions.