Angles & Coterminal Angles
Describe rotations, locate terminal sides, and recognize angles that share the same direction.
Start with a rotation
Imagine a ray turning like a pointer about a fixed pivot. An angle records that turn, including its direction, rather than just the final opening between two lines. This viewpoint will let us describe clockwise motion and turns larger than one revolution using the same language.
An angle is formed when a ray rotates about its endpoint, the vertex. The starting ray is the initial side; the final ray is the terminal side. The angle records both how far and in which direction the ray turns.
- Describe positive and negative rotations.
- Sketch angles in standard position and identify their quadrants or axes.
- Find coterminal angles, including angles larger than one full turn.
Direction and standard position
Think of an angle as a turn you could actually make with a ray. To compare two turns, we need a shared starting position; that is the purpose of standard position. As you sketch, begin with the initial side and follow the direction of rotation before drawing the terminal side.
Counterclockwise rotation is positive; clockwise rotation is negative. A full turn measures \(360^\circ\). In standard position, the vertex is at the origin and the initial side lies on the positive \(x\)-axis.
Move the angle control slowly across an axis. Notice that an angle on an axis is quadrantal: it is not inside either neighboring quadrant.
Quadrants and quadrantal angles
A location, not an amount of rotation
A quadrant is one of the four open regions between the coordinate axes. A quadrantal angle ends on an axis and therefore belongs to no quadrant.
Classifying a terminal side tells us its final location, but does not tell us how many turns were made to get there.
Now that the starting ray and direction are fixed, imagine following the turn rather than memorizing a picture. Ask where the terminal side stops. Sometimes it lies inside a quadrant; sometimes it lands exactly on an axis, which needs its own description.
| Location | Representative angles |
|---|---|
| Quadrant I | \(0^\circ<\theta<90^\circ\) |
| Quadrant II | \(90^\circ<\theta<180^\circ\) |
| Quadrant III | \(180^\circ<\theta<270^\circ\) |
| Quadrant IV | \(270^\circ<\theta<360^\circ\) |
| Positive x-axis | \(0^\circ,\ 360^\circ,\ -360^\circ\) |
| Positive y-axis | \(90^\circ,\ -270^\circ\) |
| Negative x-axis | \(180^\circ,\ -180^\circ\) |
| Negative y-axis | \(270^\circ,\ -90^\circ\) |
The interval descriptions use a representative angle between \(0^\circ\) and \(360^\circ\). Other rotations may end on exactly the same ray.
Sketching positive and negative angles
A signed angle records a direction as well as an amount of rotation. Trace that motion from the positive horizontal axis, using complete turns and familiar quarter turns as landmarks. Your sketch need not measure every degree precisely, but its terminal side should agree with the direction and quadrant of the given angle.
A positive turn
Show worked solution
Start on the positive \(x\)-axis. Rotate counterclockwise through \(50^\circ\), which is less than a right angle. The terminal side is in Quadrant I.
A clockwise turn
Show worked solution
Rotate clockwise almost half a turn. Add a full turn to compare with a positive angle:
\[-170^\circ+360^\circ=190^\circ\]Since \(180^\circ<190^\circ<270^\circ\), the terminal side is in Quadrant III.
More than one turn
Show worked solution
Make one full clockwise turn, then another quarter-turn clockwise. The terminal side lies on the negative y-axis, so this is a quadrantal angle.
Coterminal angles
A drawing shows where a turn finishes, but it may hide how many complete revolutions occurred along the way. This is why different angle measures can share a terminal side. Keep the final direction separate from the total amount of turning.
Angles in standard position are coterminal when they have the same terminal side. Adding or subtracting whole turns changes the rotation but not the final direction.
\[\theta+360^\circ k,\qquad k\in\mathbb Z\]The same direction
Show worked solution
Their difference is one full turn. One travels counterclockwise; the other travels clockwise, but both end on the same ray.
Coterminal does not mean equal as rotations. It means equal terminal directions.
Choose a representative angle
We now know how a full revolution changes the measure without changing the final direction. Let us use that freedom to choose a convenient representative. The requested interval matters: keep adding or removing complete turns until your result lies inside it, then check the endpoints carefully.
To find the unique representative \(0^\circ\le\alpha<360^\circ\), add or subtract full turns until the angle lies in that interval.
A negative angle
Show worked solution
The terminal side is in Quadrant IV.
A large positive angle
Show worked solution
The terminal side is in Quadrant III.
Several clockwise turns
Show worked solution
Five full turns bring the angle into the required interval. Its terminal side is in Quadrant III.
Use \(0^\circ\), not \(360^\circ\), when the requested interval excludes its upper endpoint.
When does a repeated turn return to the start?
Coterminal angles let us replace a long rotation by its final direction. For repeated motion, the next question is different: how many equal steps are needed to return to the starting ray? We must find the first positive number of steps whose total is a whole number of turns.
For an integer step of \(d^\circ\), return after \(n\) steps means \(nd=360k\) for an integer \(k\). The smallest positive number is \(n=360/\gcd(360,|d|)\) when \(d\ne0\). Here the greatest common divisor is the largest positive integer dividing both numbers. Reduce the fraction \(|d|/360\): its denominator is the required step count.
An indexing wheel
A machine turns a wheel \(84^\circ\) counterclockwise at every step. When does its direction first repeat the starting direction, and how many complete turns has it made?
Show worked solution
Since \(84/360=7/30\), after \(n\) steps the turn count is \(7n/30\). This is an integer exactly when \(n\) is divisible by \(30\), because \(7\) and \(30\) have no common factor. The first return is at \(30\) steps, after \(7\) full turns. Stopping after \(4\) steps would leave \(336^\circ\), close to but different from the initial direction.
Dividing coterminal angles by the same number need not preserve their terminal sides: \(0^\circ\) and \(360^\circ\) are coterminal, but their halves \(0^\circ\) and \(180^\circ\) are opposite. The difference must still be a whole turn after the operation.
Predict a return
A wheel starts on the positive x-axis and turns 120° at each step. When does it first return to its starting ray?
Practice: directions, turns, and reasoning
As you work through these questions, explain the direction of rotation before calculating a representative angle. Then check that your numerical answer matches your sketch. That small check catches sign errors much more reliably than repeating the same arithmetic.
1 · Sketch and classify
Show worked solution
Quadrant III.
2 · A clockwise rotation
Show worked solution
Quadrant III.
3 · Compare two rotations
Show worked solution
Yes. Both end in Quadrant I; their difference is three full turns.
4 · Find two more
Show worked solution
5 · Axis or quadrant?
Show worked solution
It is quadrantal, on the positive x-axis. It is not in Quadrant I.
6 · Explain the error
Show worked solution
They are coterminal angles. The second rotation contains one additional full counterclockwise turn, so their measures are different.
Additional practice: mixed exercises
Recognize standard position
For each description, decide whether the angle is in standard position and explain why.
- An angle has its vertex to the left of the origin and its initial ray points right.
- An angle has its vertex at the origin, initial ray on the positive x-axis, and terminal ray turned counterclockwise by 70 degrees.
- An angle has its vertex at the origin and its initial ray on the negative x-axis.
Show worked solution
- No. The vertex must be at the origin, even when the initial ray points right.
- Yes. Both conditions hold: vertex at the origin and initial ray on the positive x-axis.
- No. Standard position requires the positive x-axis as the initial ray.
Locate the terminal side
Sketch each angle in standard position. State its quadrant or the axis on which its terminal side lies.
- \(50^\circ\)
- \(-170^\circ\)
- \(300^\circ\)
- \(-450^\circ\)
- \(1330^\circ\)
- \(-1550^\circ\)
- \(750^\circ\)
- \(-330^\circ\)
- \(-820^\circ\)
- \(260^\circ\)
Show worked solution
- Add or subtract whole turns of \(360^\circ\): a coterminal angle is \(50^\circ\). Terminal side: Quadrant I.
- Add or subtract whole turns of \(360^\circ\): a coterminal angle is \(190^\circ\). Terminal side: Quadrant III.
- Add or subtract whole turns of \(360^\circ\): a coterminal angle is \(300^\circ\). Terminal side: Quadrant IV.
- Add or subtract whole turns of \(360^\circ\): a coterminal angle is \(270^\circ\). Terminal side: negative y-axis.
- Add or subtract whole turns of \(360^\circ\): a coterminal angle is \(250^\circ\). Terminal side: Quadrant III.
- Add or subtract whole turns of \(360^\circ\): a coterminal angle is \(250^\circ\). Terminal side: Quadrant III.
- Add or subtract whole turns of \(360^\circ\): a coterminal angle is \(30^\circ\). Terminal side: Quadrant I.
- Add or subtract whole turns of \(360^\circ\): a coterminal angle is \(30^\circ\). Terminal side: Quadrant I.
- Add or subtract whole turns of \(360^\circ\): a coterminal angle is \(260^\circ\). Terminal side: Quadrant III.
- Add or subtract whole turns of \(360^\circ\): a coterminal angle is \(260^\circ\). Terminal side: Quadrant III.
Coterminal angles
Find the coterminal angle in \([0^\circ,360^\circ)\).
- \(-25^\circ\)
- \(-30^\circ\)
- \(620^\circ\)
- \(-305^\circ\)
Show worked solution
- \(-25+360=335\), so the answer is \(335^\circ\).
- \(-30+360=330\), so the answer is \(330^\circ\).
- \(620-360=260\), so the answer is \(260^\circ\).
- \(-305+360=55\), so the answer is \(55^\circ\).
Takeaways and next lesson
- Specify direction as well as magnitude.
- Standard position fixes the vertex and initial side.
- Axes belong to quadrantal angles; quadrants exclude their boundaries.
- Use \(\theta+360^\circ k\) for all coterminal angles.
Reason about directions and repeated turns
A terminal ray records a direction, but it forgets how many turns were made. These problems ask you to separate that final direction from the actual motion. Use the angle explorer to check your prediction after reasoning on paper.
Application · A camera returns to its direction
A camera starts at \(25^\circ\) in standard position. It turns counterclockwise through \(810^\circ\), then clockwise through \(450^\circ\). Find its final representative in \([0^\circ,360^\circ)\), its net signed turn, and its total angular travel.
Hint
Keep clockwise turns negative for the net turn. Total travel adds the magnitudes instead.
Show worked solution
The final accumulated angle is \(25^\circ+360^\circ=385^\circ\), whose representative is \(25^\circ\). Total travel is \(810^\circ+450^\circ=1260^\circ\). Returning to the same direction does not imply zero movement or zero net turn.
Reasoning · Find every permitted rotation
A rotating sign must finish at \(70^\circ\). Its motor accepts rotations \(\theta\) with \(-720^\circ\le\theta\lt 720^\circ\). Starting from the positive horizontal ray, find all accepted rotations that work. Explain why your list is complete.
Hint
Write the coterminal family and solve the interval inequality for the integer turn count.
Show worked solution
The integers in this interval are \(-2,-1,0,1\), giving \(-650^\circ,-290^\circ,70^\circ,430^\circ\). Every coterminal rotation has the displayed form, and every permitted integer has been included, so no other answer is possible.
Advanced / Honors · A unique direction label
Prove that every real angle measure \(\theta\), expressed in degrees, has exactly one coterminal representative in \([0,360)\). Why would using the closed interval \([0,360]\) spoil uniqueness?
Hint
Use the integer \(k=\lfloor\theta/360\rfloor\), then compare two hypothetical representatives.
Show worked solution
By the definition of the floor, \(k\le\theta/360\lt k+1\). Thus \(\alpha=\theta-360k\) satisfies \(0\le\alpha\lt 360\) and is coterminal with \(\theta\). If \(\alpha,\beta\) are two such representatives, \(\alpha-\beta=360m\) for an integer \(m\). Their bounds imply \(-360\lt \alpha-\beta\lt 360\), so the only possible multiple is zero. Hence \(\alpha=\beta\). In the closed interval, both \(0\) and \(360\) represent the positive horizontal direction.
