This section serves as a review of the concept of ‘angle’ and the use of the degree system to measure angles. Recall that a ray is usually described as a ‘half-line’ and can be thought of as a line segment in which one of the two endpoints is pushed off infinitely distant from the other, as pictured below. The point from which the ray originates is called the initial point of the ray.

When two rays share a common initial point they form an angle and the common initial point is called the vertex of the angle. Two examples of what are commonly thought of as angles are

However, the two figures below also depict angles - albeit these are, in some sense, extreme cases. In the first case, the two rays are directly opposite each other forming what is known as a straight angle; in the second, the rays are identical so the ‘angle’ is indistinguishable from the ray itself.

The measure of an angle is a number which indicates the amount of rotation that separates the rays of the angle. There is one immediate problem with this, as pictured below.

Which amount of rotation are we attempting to quantify? What we have just discovered is that we have at least two angles described by this diagram. Clearly these two angles have different measures because one appears to represent a larger rotation than the other, so we must label them differently. In this book, we use lower case Greek letters such as \(\alpha \) (alpha), \(\beta \) (beta), \(\gamma \) (gamma) and \(\theta \) (theta) to label angles. So, for instance, we have

One system to measure angles is degree measure. Quantities measured in degrees are denoted by the symbol ‘\(^{\circ }\).’ One complete revolution as shown below is \(360^{\circ }\), and parts of a revolution are measured proportionately. Thus half of a revolution (a straight angle) measures \(\frac {1}{2} \left (360^{\circ }\right ) = 180^{\circ }\), a quarter of a revolution (a right angle) measures \(\frac {1}{4} \left (360^{\circ }\right ) = 90^{\circ }\) and so on.

Note that in the above figure, we have used the small square to denote a right angle, as is commonplace in Geometry. Recall that if an angle measures strictly between \(0^{\circ }\) and \(90^{\circ }\) it is called an acute angle and if it measures strictly between \(90^{\circ }\) and \(180^{\circ }\) it is called an obtuse angle. It is important to note that, theoretically, we can know the measure of any angle as long as we know the proportion it represents of entire revolution. For instance, the measure of an angle which represents a rotation of \(\frac {2}{3}\) of a revolution would measure \(\frac {2}{3} \left (360^{\circ }\right ) = 240^{\circ }\), the measure of an angle which constitutes only \(\frac {1}{12}\) of a revolution measures \(\frac {1}{12} \left (360^{\circ }\right ) = 30^{\circ }\) and an angle which indicates no rotation at all is measured as \(0^{\circ }\).

Using our definition of degree measure, we have that \(1^{\circ }\) represents the measure of an angle which constitutes \(\frac {1}{360}\) of a revolution. Even though it may be hard to draw, it is nonetheless not difficult to imagine an angle with measure smaller than \(1^{\circ }\). There are two ways to subdivide degrees. The first, and most familiar, is decimal degrees. For example, an angle with a measure of \(30.5^{\circ }\) would represent a rotation halfway between \(30^{\circ }\) and \(31^{\circ }\), or equivalently, \(\frac {30.5}{360} = \frac {61}{720}\) of a full rotation. This can be taken to the limit using Calculus so that measures like \(\sqrt {2}^{\, \circ }\) make sense. The second way to divide degrees is the Degree - Minute - Second (DMS) system. In this system, one degree is divided equally into sixty minutes, and in turn, each minute is divided equally into sixty seconds. In symbols, we write \(1^{\circ } = 60'\) and \(1' = 60''\), from which it follows that \(1^{\circ } = 3600''\). To convert a measure of \(42.125^{\circ }\) to the DMS system, we start by noting that \(42.125^{\circ } = 42^{\circ } + 0.125^{\circ }\). Converting the partial amount of degrees to minutes, we find \(0.125^{\circ } \left ( \frac {60'}{1^{\circ }} \right ) = 7.5' = 7' + 0.5'\). Converting the partial amount of minutes to seconds gives \(0.5' \left (\frac {60''}{1'} \right ) = 30''\). Putting it all together yields

\[ \begin{array}{rcl} 42.125^{\circ } & = & 42^{\circ } + 0.125^{\circ } \\ & = & 42^{\circ } + 7.5' \\ & = & 42^{\circ } + 7' + 0.5' \\ & = & 42^{\circ } + 7' + 30'' \\ & = & 42^{\circ } 7' 30'' \\ \end{array} \]

On the other hand, to convert \(117^{\circ }15'45''\) to decimal degrees, we first compute \(15' \left (\frac {1^{\circ }}{60'}\right ) = \frac {1}{4}^{\circ }\) and \(45'' \left (\frac {1^{\circ }}{3600''}\right ) = \frac {1}{80}^{\circ }\). Then we find

\[ \begin{array}{rcl} 117^{\circ }15'45'' & = & 117^{\circ } + 15' + 45'' \\ & = & 117^{\circ } + \frac {1}{4}^{\circ } + \frac {1}{80}^{\circ } \\ & = & \frac {9381}{80}^{\circ } \\ & = & 117.2625^{\circ } \\ \end{array} \]

Recall that two acute angles are called complementary angles if their measures add to \(90^{\circ }\). Two angles, either a pair of right angles or one acute angle and one obtuse angle, are called supplementary angles if their measures add to \(180^{\circ }\). In the diagram below, the angles \(\alpha \) and \(\beta \) are supplementary angles while the pair \(\gamma \) and \(\theta \) are complementary angles.

In practice, the distinction between the angle itself and its measure is blurred so that the sentence ‘\(\alpha \) is an angle measuring \(42^{\circ }\)’ is often abbreviated as ‘\(\alpha = 42^{\circ }\).’ It is now time for an example.

Up to this point, we have discussed only angles which measure between \(0^{\circ }\) and \(360^{\circ }\), inclusive. Ultimately, we want to use the arsenal of Algebra which we have stockpiled in Chapters IntroductiontoFunctions through SequencesandtheBinomialTheorem to not only solve geometric problems involving angles, but also to extend their applicability to other real-world phenomena. A first step in this direction is to extend our notion of ‘angle’ from merely measuring an extent of rotation to quantities which indicate an amount of rotation along with a direction. To that end, we introduce the concept of an oriented angle. As its name suggests, in an oriented angle, the direction of the rotation is important. We imagine the angle being swept out starting from an initial side and ending at a terminal side, as shown below. When the rotation is counter-clockwise from initial side to terminal side, we say that the angle is positive; when the rotation is clockwise, we say that the angle is negative.

At this point, we also extend our allowable rotations to include angles which encompass more than one revolution. For example, to sketch an angle with measure \(450^{\circ }\) we start with an initial side, rotate counter-clockwise one complete revolution (to take care of the ‘first’ \(360^{\circ }\)) then continue with an additional \(90^{\circ }\) counter-clockwise rotation, as seen below.

To further connect angles with the Algebra which has come before, we shall often overlay an angle diagram on the coordinate plane. An angle is said to be in standard position if its vertex is the origin and its initial side coincides with the positive horizontal (usually labeled as the \(x\)-) axis. Angles in standard position are classified according to where their terminal side lies. For instance, an angle in standard position whose terminal side lies in Quadrant I is called a ‘Quadrant I angle’. If the terminal side of an angle lies on one of the coordinate axes, it is called a quadrantal angle. Two angles in standard position are called coterminal if they share the same terminal side. In the figure below, \(\alpha = 120^{\circ }\) and \(\beta = -240^{\circ }\) are two coterminal Quadrant II angles drawn in standard position. Note that \(\alpha = \beta + 360^{\circ }\), or equivalently, \(\beta = \alpha - 360^{\circ }\). We leave it as an exercise to the reader to verify that coterminal angles always differ by a multiple of \(360^{\circ }\). More precisely, if \(\alpha \) and \(\beta \) are coterminal angles, then \(\beta = \alpha + 360^{\circ } \cdot k\) where \(k\) is an integer.

Note that since there are infinitely many integers, any given angle has infinitely many coterminal angles, and the reader is encouraged to plot the few sets of coterminal angles found in Example orientedcoterminaldegree to see this.

As we’ll see in Section AppRightTrig and throughout Chapter GeometricApplicationsofTrigonometry, degree measure is very popular for many applications involving geometry and modeling physical forces. In Section RadianMeasure, we’ll introduce a different method of measuring angles, radian measure, which is tied directly to arc length and is useful in other applications involving circular motion and periodic phenomenon.