Angle equivalence on the unit circle: A positive angle of 30° is coterminal with which negative angle (i.e., differs by an integer multiple of 360°)?

Introduction / Context:
In engineering mathematics and AC phasor analysis, angles that differ by integer multiples of 360° (or 2π radians) are considered coterminal because they point to the same direction on the unit circle. Being fluent with coterminal angles helps when wrapping phases into principal ranges and comparing phasors in circuit problems.

Given Data / Assumptions:

• Reference positive angle: 30°.
• We seek an equivalent negative angle that is coterminal with 30°.
• Coterminal criterion: θ_negative = 30° − 360° * k for some integer k, typically k = 1 to find the nearest negative representative.

Concept / Approach:
Angles are periodic with period 360° in degree measure. Two angles are coterminal if their difference is 360° * n, where n is any integer. To express 30° as a negative coterminal angle, subtract 360° once to land in the negative range while preserving direction on the circle.

Step-by-Step Solution:

Verification / Alternative check:
Add 360° back to −330° to confirm: −330° + 360° = 30°. This satisfies the coterminal definition directly and confirms the equivalence.

Why Other Options Are Wrong:

• −30° and −60°: These differ from 30° by only 60° or 90°, not an integer multiple of 360°.
• −180°: Opposite direction of 180°, not coterminal with 30°.

Common Pitfalls:

• Forgetting that coterminal requires adding or subtracting 360° * n, not just any angle.
• Mixing degree and radian systems without proper conversion.

Final Answer:
–330°