In geometry an angle was an opening between two rays, somewhere from $0^\circ$ to $360^\circ$. Trigonometry needs more than that, because it describes things that turn: wheels, tides, the hands of a clock. So we now think of an angle as a rotation.
Place the vertex at the origin and start a ray on the positive $x$-axis. That ray is the initial side. Turn it about the origin; where it stops is the terminal side. An angle drawn this way is in standard position.
Key idea
Counterclockwise turns are positive and clockwise turns are negative. Angles that end on the same ray are coterminal; they differ by a whole number of full turns, $360^\circ$ each.
Why it works
Measuring a turn, not just an opening, lets an angle go past $360^\circ$ or below $0^\circ$. A wheel that spins one full turn and then $40^\circ$ more has turned $400^\circ$, and it ends exactly where a $40^\circ$ turn ends. Positive is counterclockwise so that a growing angle visits Quadrants I, II, III and IV in order.
Worked examples
Worked example: A big positive angle
Find the angle between $0^\circ$ and $360^\circ$ that is coterminal with $780^\circ$.
- $780^\circ$ is more than one full turn, so remove full turns of $360^\circ$.
- $780^\circ - 360^\circ = 420^\circ$, still more than a turn.
- $420^\circ - 360^\circ = 60^\circ$.
Answer: $60^\circ$. The angle makes two full turns and then $60^\circ$ more.
Worked example: A negative angle
Draw $-135^\circ$ and name its quadrant. Then find a positive coterminal angle.
- Negative means clockwise: from the positive $x$-axis, turn clockwise past $-90^\circ$ (straight down) and $45^\circ$ farther.
- The terminal side points down and to the left: Quadrant III.
- Add a full turn: $-135^\circ + 360^\circ = 225^\circ$.
Answer: Quadrant III; $225^\circ$ is coterminal with $-135^\circ$.
Common mistake
Turning clockwise for a positive angle. Then $-90^\circ$ and $90^\circ$ get swapped. Quick check: $90^\circ$ always points straight up (the positive $y$-axis) and $-90^\circ$ straight down. Another slip is to say $400^\circ$ "isn't a real angle." It is: one full turn plus $40^\circ$.
- Quadrant I: between $0^\circ$ and $90^\circ$.
- Quadrant II: between $90^\circ$ and $180^\circ$.
- Quadrant III: between $180^\circ$ and $270^\circ$.
- Quadrant IV: between $270^\circ$ and $360^\circ$. Angles that land exactly on an axis ($0^\circ$, $90^\circ$, …) are in no quadrant.