Coterminal Angles in Trigonometry

Coterminal Angles

Introduction to Coterminal Angles

Coterminal angles are a fundamental concept in trigonometry, especially when working with angles outside the standard range of 0° to 360°. Understanding coterminal angles helps in identifying angles that share the same terminal side when drawn in standard position.

• Coterminal Angles: Two or more angles in standard position (with the same initial side) that share the same terminal side are called coterminal angles.

• To find coterminal angles, add or subtract multiples of 360° to a given angle.

• This concept is useful for expressing angles outside the typical 0° to 360° range in a more familiar form.

Key Formula:

$\theta_{\text{coterminal}} = \theta + 360^\circ \times k$

where $k$ is any integer (positive, negative, or zero).

Examples of Finding Coterminal Angles

• Example 1: Find an angle between 0° and 360° that is coterminal with 370°.

  • Subtract 360°: $370^\circ - 360^\circ = 10^\circ$

  • Answer: 10° is coterminal with 370°.

• Example 2: Find an angle between 0° and 360° that is coterminal with -150°.

  • Add 360°: $-150^\circ + 360^\circ = 210^\circ$

  • Answer: 210° is coterminal with -150°.

• Example 3: Find an angle between 0° and 360° that is coterminal with 1000°.

  • Subtract 360° repeatedly: $1000^\circ - 360^\circ = 640^\circ$; $640^\circ - 360^\circ = 280^\circ$

  • Answer: 280° is coterminal with 1000°.

Visualizing Coterminal Angles

When angles are drawn in standard position (initial side along the positive x-axis), coterminal angles will always end at the same location on the unit circle, regardless of how many full rotations are made.

Practice Problems

• Find the smallest positive angle coterminal with the given angle. Sketch the angle in standard position.

  • (A) 710°: $710^\circ - 360^\circ = 350^\circ$ (smallest positive coterminal angle is 350°)

  • (B) -37°: $-37^\circ + 360^\circ = 323^\circ$ (smallest positive coterminal angle is 323°)

  • (C) -480°: $-480^\circ + 360^\circ = -120^\circ$; $-120^\circ + 360^\circ = 240^\circ$ (smallest positive coterminal angle is 240°)

Summary Table: Coterminal Angle Calculation

Given Angle	Operation	Smallest Positive Coterminal Angle
710°	710° - 360° = 350°	350°
-37°	-37° + 360° = 323°	323°
-480°	-480° + 360° = -120°; -120° + 360° = 240°	240°

Additional info: Coterminal angles are essential for simplifying trigonometric calculations and for understanding periodicity in trigonometric functions. They are also useful in applications involving rotations, such as physics and engineering.