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Angle Measure

Trigonometric Functions

Overview Angles Degrees & Radians Coterminal Angles Arc Length & Sector Area Linear & Angular Speed

Angles

Probably the most fundamental concept of this course is that of an angle, which we can intuitively describe as the amount of rotation applied to a ray about a fixed point called a vertex. The starting position of the ray is called the initial side and the ending position is called the terminal side. If the rotation is in a counterclockwise direction, then the angle is positive. If the angle rotates in a clockwise direction, then the angle is negative. The amount of rotation in either direction is unrestricted, meaning the ray can rotate about the vertex with multiple complete revolutions.

Rotation of a ray starting at an initial side and rotating counter-clockwise to a terminal side about a vertex point at the end of the ray.

Standard Position

In order for us to talk about angles mathematically using concepts we already know, like graphs and functions, it is helpful to define a standard way to draw angles. An angle is drawn in standard position if its vertex is at the origin and the initial side is along the positive \(x\)-axis.

Which of the following illustrations gives an angle in standard position?

Image of an angle not in standard position with clockwise rotation having vertex at the origin, but both the initial and terminal sides point into the 2nd quadrant.
Image of an angle in standard position with counterclockwise rotation having vertex at the origin, initial side on the positive x-axis, and terminal side pointing into the 1st quadrant.
Image of an angle not in standard position with counterclockwise rotation having vertex at the origin, initial side parallel to the positive x-axis, and terminal side pointing into the 1st quadrant.

The direction of the rotation does not effect whether the angle is in standard position or not. The angle in the first graph is located with its vertex at the origin, but its initial side is not on the positive \(x\)-axis. Similarly, the angle in the third graph has an initial side that is parallel to the positive \(x\)-axis, but its vertex is not located at the origin. So only the angle given in the 2nd graph would be in standard position because its vertex is at the origin and its initial side is along the positive \(x\)-axis.