Oer-Trigonometry - Notes

1 CHAPTER 1 ANGLES AND THEIR MEASURE Chapter Outline 1.1 Degree Measure of Angles 1.2 Radian Measure of Angles 1.3 Applications of Radian Measure Introduction In this chapter we introduce angles in preparation for their critical role in the study of trigonometry. Beginning with the degree measure of angles in Section 1.1, we move on to radian measure in Section 1.2. The inclusion of conversion between degrees and radians leads to Section 1.3, where radian measure proves useful in solving applications of circular arc length and area, as well as linear and angular speed. Throughout this chapter, emphasis remains on angle measure and graphing. In preparation for Chapter 2, angles are graphed in standard position and coterminal angles are identified. The applications at the end of Chapter 1 will be the first of many real-world uses to occur throughout our study of trigonometry. 2 1.1 Degree Measure of Angles Learning Objectives In this section you will:  Convert between the Degree-Minute-Second system and decimal degrees.  Determine supplementary and complementary angle measures.  Graph angles in standard position.  Determine coterminal angle measures. To get started, we recall some basic definitions from geometry. A ray, often described as a half-line, is a subset of a line that contains a point P along with all points lying to one side of P. The point P from which the ray originates is called the initial point of the ray, as pictured below. P When two rays share a common initial point they form an angle and the common initial point is called the vertex of the angle. Following are two examples of angles, the first with vertex R and the second with vertex S. S R The two figures below also depict angles. 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. T P 3 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 the following pictures indicate. 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.1 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 α (alpha), β (beta), γ (gamma) and θ (theta) to label angles. So, for instance, we have   Degree Measure One commonly used system to measure angles is degree measure. Quantities measured in degrees are denoted by a small circle displayed as a superscript. One complete revolution is 360°, and parts of a revolution are measured proportionately.2 Thus half of a revolution (a straight angle) measures 1 360  180 ; a quarter of a revolution measures 14 360   90 , and so on. 2 1 2 The phrase ‘at least’ will be justified in short order. The choice of 360 is most often attributed to the Babylonians. 4 One revolution ↔ 360° 180° 90° Note that in the previous figure we have used the small square to denote a right angle, as is commonplace in geometry. If an angle measures strictly between 0° and 90° it is called an acute angle and if it measure strictly between 90° and 180° it is called an obtuse angle. We can determine the measure of any angle as long as we know the proportion it represents of an entire revolution.3 For instance, the measure of an angle which represents a rotation of 2 of a revolution would 3 measure 2 360   240 . The measure of an angle which constitutes only 121 of a revolution would 3 measure 1  360   30 . An angle which indicates no rotation at all is measured as 0°. 12 240° 3 This is how a protractor is graded. 30° 0° 5 Using our definition of degree measure, 1° represents the measure of an angle which constitutes 1 of a 360 revolution. Even though it may be hard to draw, it is nonetheless not difficult to imagine an angle with measure smaller than 1°. There are two ways to subdivide degrees. 1. The first, and most familiar, is decimal degrees. For example, an angle with a measure of 30.5° would represent a rotation halfway between 30° and 31° or, equivalently 30.5 61  of a full 360 720 rotation. 2. The second way to divide degrees is the Degree-Minute-Second (DMS) system, used in surveying, global positioning and other applications requiring measurements of longitude and latitude. In the DMS 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  60' and 1'  60" , from which it follows that 1  3600" . EXAMPLE 1.1.1. Convert 42.125° to the DMS system. SOLUTION. To convert 42.125° to the DMS system, we first note that 42.125  42  0.125 . Converting the partial amount of degrees to minutes, we find  60'  0.125    7.5'  1   7' 0.5'. Next, converting the partial amount of minutes to seconds gives  60"   0.5'  30".  1'  The result is 42.125  42  0.125  42  7.5'  42  7' 0.5'  42  7' 30"  42 7'30". 6 EXAMPLE 1.1.2. Convert 117°15'45" to decimal degrees. SOLUTION. To convert 117°15'45" to decimal degrees, we first compute  1 1 15'  60'   4  0.25 and  1 1 45"   3600"  80  0.0125 . Then we find 117 15'45"  117  15' 45"  117  0.25  0.0125  117.2625 Supplementary and Complementary Angles Recall that two acute angles are called complementary angles if their measures add to 90°. 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°. In the diagrams below, the angles α and β are supplementary angles while the pair γ and θ are complementary angles.     In practice, the distinction between the angle itself and its measure is blurred so that the statement ‘α is an angle measuring 42°’ is often abbreviated as ‘  42 ’. EXAMPLE 1.1.3. Let   111.371 and   37 28'17" . 1. Sketch α and β. 7 2. Find a supplementary angle for α. 3. Find a complementary angle for β. SOLUTION. 1. To sketch   111.371 , we first note that 90    180 . If we divide this range in half, we observe that 90    135 . After one more division, we get 90    112.5 . This gives us a fairly good estimate for α, as shown in the figure.  For   37 28'17" , converting to decimal degrees results in approximately 37.47°. We find 0    90 . After dividing this range in half, we get 0    45 , followed by 22.5    45 , and lastly 33.75    45 .  2. To find a supplementary angle for   111.371 , we seek an angle θ so that    180 . We get  180   180 111.371  68.629 . 3. To find a complementary angle for   37 28'17" , we seek an angle γ so that    90 . We get  90  90 37 28'17" . While we could reach for the calculator to obtain an approximate answer, we choose instead to do a bit of sexagesimal4 arithmetic. We first rewrite 4 This is a base-60 system. 8 90  90 0'0"  89 60'0"  89 59'60". In essence, we are ‘borrowing’ 1°=60' from the degree place, and then borrowing 1'=60" from the minutes place.5 This yields  90 37 28'17" 89 59'60" 37 28'17" 52 31'43". Oriented Angles Up to this point, we have discussed only angles which measure between 0° and 360°, inclusive. Ultimately, we want to extend their applicability to other real-world phenomena. A first step in this direction is to 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 the angle is negative. rm ina Te ide lS ina rm Te lS ide Initial Side Initial Side A positive angle, 60° A negative angle, –60° We also extend our allowable rotations to include angles which encompass more than one revolution. For example, to sketch an angle with measure 450° we start with an initial side, rotate counter-clockwise one complete revolution (to take care of the first 360°), then continue with an additional 90° counterclockwise rotation, as seen below. 5 This is the exact kind of borrowing often taught in elementary school when trying to find 300–125. The difference is that a base ten system is used to find 300–125; here, it is base sixty. 9 450° Standard Position 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 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.6 In the following figure,   120 and   240 are two coterminal Quadrant II angles drawn in standard position. Note that     360 , or equivalently     360 . We leave it as an exercise for the reader to verify that coterminal angles always differ by a multiple of 360°.7 More precisely, if α and β are coterminal angles, then    360  k where k is an integer.8 y   120 x    240 6 Note that by being in standard position they automatically share the same initial side which is the positive x-axis. It is worth noting that all of the pathologies of Analytic Trigonometry result from this innocuous fact. 8 Recall that this means k = 0, ±1, ±2, ···. 7 10 EXAMPLE 1.1.4. Graph each of the (oriented) angles below in standard position and classify them according to where their terminal side lies. Find three coterminal angles, at least one of which is positive and one of which is negative. 1.   60 2.    225 3.   540 4.   750 SOLUTION. y 1. To graph   60 , we draw an angle with its initial side on the positive x-axis and rotate counter-clockwise 60 360 1 of a 6   60 revolution. We see that α is a Quadrant I angle. x To find angles which are coterminal, we look for angles θ of the form     360  k for some integer k.  When k  1, we get   60  360  420 .  Substituting k  1 gives   60  360  300 .  If we let k  2 , we get   60  720  780 . 2. Since    225 is negative, we start at the positive x-axis and rotate clockwise 225 360 y 5 of a revolution. We see that β is a 8 x Quadrant II angle. To find coterminal angles, we proceed as before and compute   225   225  360  k for integer values of k. Letting k  1 , k  1 and k  2 , we find 135°, –585° and 495° are all coterminal with –225°. 3. Since   540 is positive, we rotate counter-clockwise from the y   540 positive x-axis. One full revolution accounts for 360°, with 180°, or half of a revolution, remaining. Since the terminal side of γ lies on the negative x-axis, γ is a quadrantal angle. All angles coterminal with γ are of the form   540  360  k , where k is an integer. Working through the arithmetic, we find three such angles: 900°, 180° and –180°. x