Angles, or Rotations?

UNH Mathematics Center

What's the difference between an angle and a rotation?

• We usually think of angles in the context of a triangle. The memory-aid SOHCAHTOA (''opposite over hypotenuse, adjacent over hypotenuse,'' etc) certainly makes us think of a right triangle! Of course, angles in a triangle are limited in size. A right triangle's largest angle is its right angle; the largest angle in any triangle must be less than two right angles.

  But the trig functions can be extended so that they apply to ''angles'' larger than the ones that can be in triangles. We will eventually think of the trig functions as applying to numbers - even negative numbers - that sometimes don't even need to be interpreted as angle measures. We can extend the trig functions in this way if we think of their input numbers as measuring rotations, rather than just angles that we would find in a triangle.

• We usually picture a rotation as a change in direction. For instance, a rotating vector has a starting direction; and then it turns (one way or the other) until it points in a new direction.

  Think of a weathervane swinging in the wind. A gust of wind might start it spinning, so that it moves just a bit and stops; or it might spin several times and then stop ...

  

  or it might spin in the reverse direction and then stop. Even with the very same initial and final positions, the rotations the wind pushes the weathervane through could be very different.

• Often when we talk about an angle we are describing the ''spread'' between the angle's two sides. Although we might picture an angle as static and unchanging, a rotation implies movement - the ''spin'' that transformed the vector's initial position to a later position.

Describing a rotation involves knowing:

• Where was the vector pointing when the rotation began?
• Where was the vector pointing at the end of the rotation?
• In which direction was the rotation? (With the sun? Against the sun?)
• Did the vector make any complete turns as part of its rotation?

Any measurement system - typically degrees or radians - that we would use for angles, can be adapted for use with rotations. When we use a measurement system with rotations, the values we get can be larger than those we would expect as angle-measurements. They can also be negative.

Taking a trig function's argument to be the measure of a rotation expands the trig function's domain.