Sketching angles in standard position
Although an angle can be drawn at any position and in any orientation in two-dimensional
space, in trigonometry it is often convenient to draw an angle on a coordinate system (an
origin, and a pair of coordinate axes that are perpendicular to each other). We think of one
side of the angle as the initial side and the other side of the angle as the terminal side.
When we draw the angle in what is known as standard position, we put its vertex at the
origin, and we draw the initial side of the angle along the positive horizontal axis. Then we
think of getting from the initial side of the angle to the terminal side of the angle by rotating
the initial side about the origin until it coincides with the terminal side. The amount of that
rotation corresponds to the measure of the angle.
Now you may be wondering what happens if one person thinks of making the rotation in
the clockwise direction and someone else thinks of making the rotation in the
counterclockwise direction. Doesn't that mean that the angle has two different measures?
To deal with that potential source of confusion, we indicate the measure of any angle by a
real number: the size of the angle (which is nonnegative) and (if the size is nonzero) the
direction of the angle (indicated by including a minus sign if the rotation described above
is in the clockwise direction).
If your intended angle is the one whose terminal side is reached by rotating the initial side
in the counterclockwise direction, the measure of the angle is positive.
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