Chapter 15 Measuring an Angle So far, the equations we have studied have an algebraic Cosmo S character, involving the variables x and y, arithmetic op- motors around erations and maybe extraction of roots. Restricting our Q the attention to such equations would limit our ability to de- circle R scribe certain natural phenomena. An important example P 20 feet involves understanding motion around a circle, and it can be motivated by analyzing a very simple scenario: Cosmo the dog, tied by a 20 foot long tether to a post, begins Figure 15.1: Cosmo the dog walking around a circle. A number of very natural ques- walking a circular path. tions arise: Natural Questions 15.0.1. How can we measure the angles ∠SPR, ∠QPR, and ∠QPS? How can we measure the arc lengths arc(RS), arc(SQ) and arc(RQ)? How can we measure the rate Cosmo is moving around the circle? If we know how to measure angles, can we compute the coordinates of R, S, and Q? Turning this around, if we know how to compute the coordinates of R, S, and Q, can we then measure the angles ∠SPR, ∠QPR, and ∠QPS ? Finally, how can we specify the direction Cosmo is traveling? We will answer all of these questions and see how the theory which evolves can be applied to a variety of problems. The definition and basic properties of the circular functions will emerge as a central theme in this Chapter. The full problem-solving power of these functions will become apparent in our discussion of sinusoidal functions in Chapter 19. The xy-coordinate system is well equipped to study straight line mo- tion between two locations. For problems of this sort, the important tool is the distance formula. However, as Cosmo has illustrated, not all two-dimensional motion is along a straight line. In this section, we will describe how to calculate length along a circular arc, which requires a quick review of angle measurement. 191 192 CHAPTER 15. MEASURING AN ANGLE 15.1 Standard and Central Angles An angle is the union of two rays emanating from a common point called the vertex of the angle. A typical angle can be dynamically generated by rotating a single ray from one position to another, sweeping counter- clockwise or clockwise: See Figure 15.2. We often insert a curved arrow to indicate the direction in which we are sweeping out the angle. The ray ℓ1 is called the initial side and ℓ2 the terminal side of the angle ∠AOB. (terminal side) l2 B SWEEP CLOCKWISE vertex vertex A (initial side) l1 l1 l1 O A O A (initial side) O (terminal side) START SWEEP COUNTERCLOCKWISE B l2 Figure 15.2: Angle ∠AOB. Working with angles, we need to agree on a standard frame of refer- ence for viewing them. Within the usual xy-coordinate system, imagine a model of ∠AOB in Figure 15.2 constructed from two pieces of rigid wire, welded at the vertex. Sliding this model around inside the xy-plane will not distort its shape, only its position relative to the coordinate axis. So, we can slide the angle into position so that the initial side is coincident with the positive x-axis and the vertex is the origin. Whenever we do this, we say the angle is in standard position. Once an angle is in stan- dard position, we can construct a circle centered at the origin and view our standard angle as cutting out a particular “pie shaped wedge” of the corresponding disc. Notice, the shaded regions in Figure 15.3 depend on whether we sweep the angle counterclockwise or clockwise from the initial side. The portion of the “pie wedge” along the circle edge, which is an arc, is called the arc subtended by the angle. We can keep track of this arc using the notation arc(AB). A central angle is any angle with vertex at the center of a circle, but its initial side may or may not be the positive x-axis. For example, ∠QPS in the Figure beginning this Chapter is a central angle which is not in standard position. 15.2. AN ANALOGY 193 y-axis l2 l2 y-axis B B arc subtended arc(AB) arc(AB) vertex vertex x-axis x-axis O l1 O l A A 1 COUNTERCLOCKWISE CLOCKWISE Figure 15.3: Standard angles and arcs. 15.2 An Analogy To measure the dimensions of a box you would use a ruler. In other words, you use an instrument (the ruler) as a standard against which you measure the box. The ruler would most likely be divided up into either English units (inches) or metric units (centimeters), so we could express the dimensions in a couple of different ways, depending on the units desired. By analogy, to measure the size of an angle, we need a standard against which any angle can be compared. In this section, we will de- scribe two standards commonly used: the degree method and the radian method of angle measurement. The key idea is this: Beginning with a circular region, describe how to construct a “basic” pie shaped wedge whose interior angle becomes the standard unit of angle measurement. 15.3 Degree Method Begin by drawing a circle of radius r, call it Cr, centered at the origin. Divide this circle into 360 equal sized pie shaped wedges, beginning with the the point (r,0) on the circle; i.e. the place where the circle crosses the x-axis. We will refer to the arcs along the outside edges of these wedges as one-degree arcs. Why 360 equal sized arcs? The reason for doing so is historically tied to the fact that the ancient Babylonians did so as they developed their study of astronomy. (There is actually an alternate system based on dividing the circular region into 400 equal sized wedges.) Any central angle which subtends one of these 360 equal sized arcs is 194 CHAPTER 15. MEASURING AN ANGLE circle Cr etc. a total of 360 this angle is equal sized r DEFINED to pie shaped typical have measure wedges inside (r,0) wedge 1 degree this disk r etc. ***NOT TO SCALE*** Figure 15.4: Wedges as 1◦ arcs. defined to have a measure of one degree, denoted 1◦. We can now use one-degree arcs to measure any angle: Begin by slid- ing the angle ∠AOB into standard central position, as in Figure 15.3. Piece together consecutive one-degree arcs in a counterclockwise or clock- wise direction, beginning from the initial side and working toward the terminal side, approximating the angle ∠AOB to the nearest degree. If we are allowed to divide a one-degree arc into a fractional portion, then we could precisely determine the number m of one-degree arcs which consecutively fit together into the given arc. If we are sweeping counter- clockwise from the initial side of the angle, m is defined to be the degree measure of the angle. If we sweep in a clockwise direction, then −m is defined to be the degree measure of the angle. So, in Figure 15.3, the left-hand angle has positive degree measure while the right-hand angle has negative degree measure. Simple examples would be angles like the ones in Figure 15.5. Notice, with our conventions, the rays determining an angle with mea- sure −135◦ sit inside the circle in the same position as those for an angle of measure 225◦; the minus sign keeps track of sweeping the positive x-axis clockwise (rather than counterclockwise). We can further divide a one-degree arc into 60 equal arcs, each called a one minute arc. Each one-minute arc can be further divided into 60 equal arcs, each called a one second arc. This then leads to angle mea- sures of one minute, denoted 1′ and one second, denoted 1′′: 1◦ = 60 minutes = 3600 seconds. 15.3. DEGREE METHOD 195 180◦ 90◦ 270◦ 45◦ 315◦ −135◦ Figure 15.5: Examples of common angles. For example, an angle of measure θ = 5 degrees 23 minutes 18 sec- onds is usually denoted 5◦ 23′ 18′′. We could express this as a decimal of degrees: In degrees! 23 18 ◦ 5◦ 23′ 18′′ = 5 + + 60 3600 ← = 5.3883◦. As another example, suppose we have an angle with measure 75.456◦ and we wish to convert this into degree/minute/second units. First, since 75.456◦ = 75◦ +0.456◦, we need to write 0.456◦ in minutes by the calculation: minutes 0.456 degree 60 = 27.36′. × degree This tells us that 75.456◦ = 75◦27.36′ = 75◦27′ + 0.36′. Now we need to write 0.36′ in seconds via the calculation: 0.36 minutes 60 seconds/minute = 21.6′′. × In other words, 75.456◦ = 75◦ 27′ 21.6′′. Degree measurement of an angle is very closely tied to direction in the plane, explaining its use in map navigation. With some additional work, it is also possible to relate degree measure and lengths of circular arcs. To do this carefully, first go back to Figure 15.3 and recall the situation 196 CHAPTER 15. MEASURING AN ANGLE where an arc arc(AB) is subtended by the central angle ∠AOB. In this situation, the arc length of arc(AB), commonly denoted by the letter s, is the distance from A to B computed along the circular arc; keep in mind, this is NOT the same as the straight line distance between the points A and B. For example, consider the six angles pictured above, of measures 90◦, 180◦, 270◦, 45◦, −135◦, and 315◦.