---
title: "Coterminal Angles — AP Precalc Definition & Exam Guide"
description: "Coterminal angles share a terminal ray and differ by full revolutions (2π or 360°). They explain why sine and cosine repeat, central to AP Precalc Unit 3."
canonical: "https://fiveable.me/ap-pre-calc/key-terms/coterminal-angles"
type: "key-term"
subject: "AP Pre-Calculus"
unit: "Unit 3"
---

# Coterminal Angles — AP Precalc Definition & Exam Guide

## Definition

Coterminal angles are angles in standard position that share the same terminal ray, differing by an integer number of full revolutions (multiples of 2π radians or 360°). Because they land in the same spot on the unit circle, coterminal angles have identical sine, cosine, and tangent values.

## What It Is

Coterminal angles are angles in [standard position](/ap-pre-calc/key-terms/standard-position "fv-autolink") (vertex at the [origin](/ap-pre-calc/unit-3/polar-function-graphs/study-guide/4Con24QzKXI6SrwldHmX "fv-autolink"), initial ray on the positive x-axis) whose terminal rays land in exactly the same place. The CED puts it precisely: angles in standard position that share a terminal ray differ by an integer number of revolutions. In radians, that means adding or subtracting multiples of 2π. In degrees, multiples of 360°.

Picture spinning a dial. If you rotate it π/3, or π/3 plus one full extra spin (π/3 + 2π), or π/3 minus two full spins, the dial points the same [direction](/ap-pre-calc/unit-4/vectors/study-guide/E38atN4oigqKq7in "fv-autolink") every time. Those are all coterminal angles. The angle measures are different numbers, but the geometry is identical. Negative angles work too, since a negative measure just means you rotated clockwise instead of counterclockwise. For example, -π/2 and 3π/2 are coterminal because they differ by exactly 2π.

## Why It Matters

Coterminal angles live in Topic 3.2 (Sine, Cosine, and Tangent) in [Unit 3](/ap-pre-calc/unit-3 "fv-autolink"): Trigonometric and Polar Functions, supporting learning objective [AP Pre Calc](/ap-pre-calc "fv-autolink") 3.2.A, which asks you to determine sine, cosine, and tangent using the unit circle. Here's why they matter beyond the definition. Sine, cosine, and tangent depend only on where the terminal ray intersects the unit circle, so coterminal angles produce identical trig values. That single fact is the seed of periodicity. Sine and cosine repeat every 2π precisely because every angle is coterminal with infinitely many others. When Unit 3 later asks you to graph sinusoidal functions, find all solutions to a trig equation, or work with polar coordinates where (r, θ) and (r, θ + 2π) name the same point, you're cashing in on coterminal angles every time.

## Connections

### [Standard Position (Unit 3)](/ap-pre-calc/key-terms/standard-position)

Coterminal angles only make sense once angles are in standard position. With every angle anchored at the origin and starting from the positive x-axis, the only thing that distinguishes two angles is [rotation](/ap-pre-calc/unit-4/matrices-as-functions/study-guide/5YRNj78FIP4lmMi9 "fv-autolink"), so angles that differ by full spins become indistinguishable.

### [Radian Measure (Unit 3)](/ap-pre-calc/key-terms/radian-measure)

One full revolution is 2π [radians](/ap-pre-calc/key-terms/radians "fv-autolink") because the arc length of a full circle is 2πr. That's why the coterminal rule is 'add or subtract multiples of 2π' in radians but 'multiples of 360' in degrees. Same idea, different units.

### Periodicity of Sine and Cosine (Unit 3)

The identity sin(θ + 2π) = sin(θ) is just the coterminal-angle fact written as an equation. When you graph y = sin(x) and see the wave repeat every 2π, you're looking at infinitely many coterminal angles producing the same [output](/ap-pre-calc/unit-1/change-tandem/study-guide/eQFiTo22fpkDFsnj "fv-autolink").

### Polar Coordinates (Unit 3)

In polar form, the point (r, θ) is the same as (r, θ + 2π). That non-uniqueness, which trips people up when solving polar problems, is coterminal angles showing up in a new costume.

## On the AP Exam

This term is multiple-choice territory, and the questions are usually fast points if you know the rule. Expect stems like 'how many degrees do two coterminal angles differ by?' (any integer multiple of 360°) or 'an angle has its vertex at the origin and initial side on the positive x-axis; which term describes it?' (standard position, the setup that makes coterminality meaningful). The most exam-relevant move is comparing trig values. A question might give you θ₁ = π/3 and θ₂ = π/3 + 2π and ask how their sine and cosine values compare. The answer is that they're identical, because both terminal rays hit the unit circle at the same point. You'll also use coterminal reasoning silently whenever you evaluate something like sin(13π/6) by rewriting it as sin(π/6), or when you find all solutions to a trig equation by tacking on '+ 2πk' to your answer.

## coterminal angles vs Reference angles

Coterminal angles are different angle measures pointing the same direction (like π/3 and π/3 + 2π). A reference angle is the acute angle between a terminal ray and the x-axis, used to relate any angle to a first-quadrant angle. Coterminal angles have identical trig values; an angle and its reference angle have trig values that match only in absolute value, with signs depending on the quadrant. Quick check: coterminal is about adding full spins, reference is about folding into Quadrant I.

## Key Takeaways

- Coterminal angles are angles in standard position that share the same terminal ray and differ by an integer number of full revolutions.
- In radians, coterminal angles differ by multiples of 2π; in degrees, they differ by multiples of 360°.
- Coterminal angles have identical sine, cosine, and tangent values because their terminal rays intersect the unit circle at the exact same point.
- Every angle has infinitely many coterminal angles, including negative ones, since you can spin extra full revolutions in either direction.
- Coterminal angles are the reason sine and cosine are periodic with period 2π, which is why trig equation solutions get written with '+ 2πk'.
- To evaluate trig functions of large or negative angles, add or subtract 2π until you land on a familiar unit circle angle, then read off the values.

## FAQs

### What are coterminal angles in AP Precalc?

Coterminal angles are angles in standard position that share the same terminal ray, differing by an integer number of full revolutions (multiples of 2π radians or 360°). For example, π/3, π/3 + 2π, and π/3 − 2π are all coterminal.

### Do coterminal angles have the same sine and cosine?

Yes, always. Their terminal rays hit the unit circle at the same point, so sine, cosine, and tangent are identical. So sin(π/3 + 2π) equals sin(π/3) exactly, with no sign change.

### How is a coterminal angle different from a reference angle?

Coterminal angles point the same direction and differ by full spins of 2π, so their trig values are identical. A reference angle is the acute angle between the terminal ray and the x-axis, and trig values only match it in absolute value, with the sign set by the quadrant.

### How do you find a coterminal angle?

Add or subtract 2π (in radians) or 360° (in degrees) as many times as you want. To find the coterminal angle between 0 and 2π for something like 13π/6, subtract 2π to get π/6... wait, 13π/6 is already in range, but 13π/6 − 2π = π/6 shows the matching first-revolution angle for 13π/6 + 2π. The process is just repeated subtraction until you land in [0, 2π).

### Can a coterminal angle be negative?

Yes. Negative measures just mean clockwise rotation, so −π/2 and 3π/2 are coterminal because they differ by exactly 2π. Every angle has infinitely many coterminal angles, both positive and negative.

## Related Study Guides

- [3.2 Sine, Cosine, and Tangent](/ap-pre-calc/unit-3/sine-cosine-tangent/study-guide/6r53DIMsbdFLUXFo)

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