Positive and Negative Angles

An angle is the amount of rotation of a ray about its fixed endpoint,

carrying it from an initial position to a terminal position. The fixed endpoint is

called the vertex of the angle, the ray in its starting position is the initial side,

and the ray in its final position is the terminal side.

Sign convention. If the rotation is anticlockwise, the angle described is taken as

positive; if the rotation is clockwise, the angle is taken as negative. This is the

single convention that lets an angle take any real value, not merely a value between

$0^\circ$ and $360^\circ$ -- the ray may be rotated through more than one complete

revolution, and the direction of rotation, not just the final position of the terminal side,

is part of what the angle records.

Standard position. An angle is said to be in standard position when its vertex is placed

at the origin of a coordinate plane and its initial side is placed along the positive x-axis.

Every angle used in this chapter is understood to be in standard position unless stated

otherwise, so that "the angle $\theta$" and "the terminal side reached by rotating the

positive x-axis through $\theta$" mean the same thing.

Measuring more than one revolution. One complete anticlockwise revolution is described as

an angle of $360^\circ$ (or $2\pi$ radians, defined in the next section). A further rotation

beyond a full turn keeps adding to the angle, so an angle of $400^\circ$, for instance, means

one full revolution ($360^\circ$) plus a further $40^\circ$ of rotation -- the terminal side

ends up in exactly the same place as for $40^\circ$, but the angle itself is a different,

larger number, because it also records the extra revolution.

Coterminal angles. Two angles that share the same initial side and the same terminal side

-- but that may differ in the number of revolutions or the direction of rotation used to reach

it -- are called coterminal angles. For example, $30^\circ$, $390^\circ$ ($=30^\circ+360^\circ$)

and $-330^\circ$ ($=30^\circ-360^\circ$) are all coterminal: each rotation ends at the same

terminal side, obtained by adding or subtracting whole multiples of $360^\circ$ from $30^\circ$.

In general, $\theta$ and $\theta + n\cdot360^\circ$ (or, in radians, $\theta$ and

$\theta+2n\pi$) are coterminal for any integer $n$. This fact -- that adding a full revolution

never changes the terminal side -- is exactly what makes every trigonometric function of

$\theta$ periodic, a property proved formally once the functions themselves are defined on the

unit circle (Section 3) and is central to the very idea of a general solution of a

trigonometric equation (Section 10): once one solution is found, every coterminal angle

obtained by adding a whole number of revolutions is automatically a solution too.

Example. An angle of $-90^\circ$ has its terminal side along the negative y-axis, reached

by a quarter-turn clockwise from the positive x-axis; an angle of $+270^\circ$ has exactly the

same terminal side, reached instead by three-quarters of a turn anticlockwise. The two angles

are coterminal, but $-90^\circ$ and $270^\circ$ are different values because they record

different amounts and directions of rotation.