Angles and The Unit Circle


  • Definition of Angle

  • The Circle: a radius and an angle

  • Dimensions and Units for Measuring Angles

    Definition of Angle

    An angle is probably best understood as a direction. If you imagine yourself pointing at an object you usually end up pointing in a particular direction, and we often talk about something being at a steep or shallow angle. Thus, to give a very basic definition of angle, one could say that it is a quantity that specifies direction in space. Obviously, to point in a direction requires for there to be more than one possible direction, and so a point or all points along a single straight line cannot have angles associated with them. The notion and concept of an angle becomes useful when there is more than one possible direction, and in physics and astronomy this usually means: in plane geomtery (2-D) and in space (3-D).

    The Circle in terms of A Radius and an Angle

    The circle is defined to be a closed geometric figure in which all points on it are at a constant distance (called a the radius r) away from a common center - to review the definition see the section on Circles and Triangles and More. Suppose not that one wanted to specify the location of an object that is moving in a circlular path, i.e., it is going around a center at a constant distance r (this is closely approximated with the motion of the Earth around the Sun, for instance). In order to specify the location of the object one would have to say how far away it is from the center, but is this all? Consider the following image:

    In order to specify where the object actually is at any given time one would have to also specify where on the circlular path it is. This can be done easily in terms of giving an angle relative to a fixed point (or a reference point) which we can call the position of "zero degrees". Thus, in the above picture we could say that the object is a distance r=10.0 centimeters away from the center and, at the moment shown, it is at an angle of q=45 from the reference location (indicated by the dashed line).

    Thus, points on a circle can be specified by giving a radius and an angle. This is significant from the point of view of trigonometric relations and specification of location of objects on a plane surface. This notion is also exteremely important in astronomy because on many occasions one is looking at objects that roughly travel in circular paths and it is important to be able to give a basic description of the motion and location of the object and how it changes with time.

    Basic Dimentions and Units for Measuring Angles

    If an angle is direction then it is a physically observable quantity, and like any other physical quantity it must be measurable (see secion on units for a review of measurable quantities).

    Measuring Angles - Degrees

    Angles can be measured in many ways since there are different ways of specifying what one means by direction. To start howerver, let us imagine a circle and see how it is defined. A circle is completed by one revolution. We define this one revolution to be equaivalent to 360 degrees (written as 360).

    Thus, one degree is equal to 1/360 th of a full rotation, as shown in figure 2 above. This is often called the one degree arc and is basically the "arc" portion of a circle that is subtended by one degree.

    The degree can further be divided into smaller units, and this is usually done in astronomy because one is usually looking at very distant objects and the measured angles on the sky tend to be extremely small (orders of magnitude smaller than the one degree). The arc minutes of arc and the seconds of arc are the common ways to divide up a degree, and they are defined as follows:

  • There are 60 minutes of arc in one degree. (The shorthand for arcminute is the single prime ('): we can write 3 arcminutes as 3'.) Therefore there are 360 × 60 = 21,600 arcminutes in a full circle.

  • There are 60 seconds of arc in one arcminute. (The shorthand for arcsecond is the double prime ("): we can write 3 arcseconds as 3".) Therefore there are 21,600 × 60 = 1,296,000 arcseconds in a full circle.

    Other Units for Angles - Radians

    We sometimes express angles in units of radians instead of degrees. If we were to take the radius (length R) of a circle and bend it so that it conformed to a portion of the circumference of the same circle, the angle subtended by that radius is defined to be an angle of one radian.

    Since the circumference of a circle has a total length of 2pR, we can fit exactly 2p radii along the circumference; thus, a full 360 circle is equal to an angle of 2p radians. In other words, an angle in radians equals the subtended arclength of a circle divided by the radius of that circle. If we imagine a unit circle (where the radius = 1 unit in length), then an angle in radians numerically equals the actual curved distance along the portion of its circumference "cut" by the angle. The conversion between radians and degrees is:

    1 radian = 360/2p = 57.3

    1 = 2p radiansv/360 = 0.017453 radian