Angle Measure
Angle measure refers to the amount of rotation between two rays that share a common endpoint, known as the vertex. It is typically measured in degrees, with a full rotation around a point equaling 360 degrees. Angles can be classified based on their measure, such as acute (less than 90 degrees), right (exactly 90 degrees), obtuse (greater than 90 degrees but less than 180 degrees), and straight (exactly 180 degrees).
4 Key excerpts on "Angle Measure"
- Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...STUDY UNIT III TRIGONOMETRIC ANALYSIS 9 TRIGONOMETRY WHAT YOU WILL LEARN Trigonometry means “measurement of triangles.” The study of trigonometry arose from the ancient need to understand the relationships between the sides and angles of triangles. With the development of calculus, trigonometry progressed from the study of ratios within right triangles to trigonometric functions that could be used to better represent the circular and repeating patterns of behavior that characterize a wide range of physical phenomena in the real world. This chapter progresses from considering acute angles in right triangles to a more general view of angles as rotations about the origin in the coordinate plane. By fixing the vertex of such an angle at the origin and keeping one side of the angle aligned with the positive x- axis, we can give meaning to trigonometric functions of angles greater than 90° and less than 0°. LESSONS IN CHAPTER 9 • Lesson 9-1: Degree and Radian Measures • Lesson 9-2: Right-Triangle Trigonometry • Lesson 9-3: The General Angle • Lesson 9-4: Working with Trigonometric Functions • Lesson 9-5: Trigonometric Functions of Special Angles Lesson 9-1: Degree and Radian Measures KEY IDEAS Angle Measures can be expressed in units of degrees or in real-number units called radians. Degrees are based on fractional parts of a circular revolution. Radian measure compares the length of an arc that a central angle of a circle cuts off to the radius of the circle. The Greek letter θ (theta) is commonly used to represent an angle of unknown measure. MEASURING ANGLES IN DEGREES AND MINUTES One degree, denoted as 1°, is of one complete revolution about a fixed point. Each of the 60 equal parts of a degree is called a minute. The notation 28°30′ is read as “28 degrees, 30 minutes.” Since 60 minutes is equivalent to 1 degree, dividing 30 minutes by 60 changes 30 minutes to a fractional part of a degree...
eBook - ePub- Surinder Virdi, Roy Baker, Narinder Kaur Virdi(Authors)
- 2014(Publication Date)
- Routledge(Publisher)
...CHAPTER 10 Geometry Learning outcomes: (a) Identify the different types of angles, triangles and quadrilaterals (b) Find angles in triangles, quadrilaterals and other geometrical constructions (c) Use Pythagoras’ theorem to determine diagonals in quadrilaterals and sides of right-angled triangles (d) Calculate the circumference of a circle 10.1 Angles When two straight lines meet at a point an angle is formed, as shown in Figure 10.1. There are two ways in which an angle can be denoted, i.e. either ∠ CAB or ∠ A. Figure 10.1 The size of an angle depends on the amount of rotation between two straight lines, as illustrated in Figure 10.2. Angles are usually measured in degrees, but they can also be measured in radians. A degree, defined as of a complete revolution, is easier to understand and use as compared to the radian. Figure 10.2 shows that the rotation of line AB makes (a) revolution or 90, (b) revolution or 180, (c) revolution or270° and (d) a complete revolution or 360°. Figure 10.2 For accurate measurement of an angle a degree is further divided into minutes and seconds. There are 60 minutes in a degree and 60 seconds in a minute. This method is known as the sexagesimal system: 60 minutes (60′) = 1 degree 60 seconds (60″) = 1 minute (1′) The radian is also used as a unit for measuring angles. The following conversion factors may be used to convert degrees into radians and vice versa. 1 radian = 57.30° (correct to 2 d.p.) π radians = 180° (π = 3.14159; correct to 5 d.p.) 2π radians = 360° Example 10.1 Convert: (a) 20°15′25″ into degrees (decimal measure) (b) 32.66° into degrees, minutes and seconds. (c) 60°25′45″ into radians. Solution: (a) The conversion of 15′25″ into degree involves two steps. The first step is to change 15′25″ into seconds, and the second to convert seconds into a degree...
eBook - ePub- Douglas K. Brumbaugh, Peggy L. Moch, MaryE Wilkinson(Authors)
- 2004(Publication Date)
- Routledge(Publisher)
...Sometimes, for the sake of convenience, line segments are used as sides of an angle, but you should understand that any line segment is a part of a ray. The best way to name an angle is to use three points in order—first a point on one leg, then the vertex point, and then a point on the other leg (the vertex letter is always the middle of the three points defining the angle). The symbol for an angle looks like a tiny angle (∠) and sometimes it has a tiny arc drawn across it (). The sides of an angle are rays, so we need not worry about sketching them any specific length. This has implications for our use of the word “congruent” and the symbol that means congruent (). Congruent is a very strong word that means figures must be exactly the same shape and exactly the same size. The sides of an angle are rays, so they extend forever no matter how long we make them appear to be. This means that we need only measure the rotations of two angles to decide if they are congruent; in Fig. 4.9, ∠DEF ∠HIJ because they have the same measure, even though we have not sketched the rays to look the same. Fig. 4.9. Because the legs of an angle are rays, extending forever, an angle divides the plane into three distinct parts, the set of all points that are inside the legs, the set of all points that are on the legs, and the set of all points that are outside the legs. This may not be obvious when you look at an angle, because we draw only a tiny part of each ray. Figure 4.10 shows interior points, like P, are between the rays, exterior points, like Q, are outside the sides, and angle points, like D, E, and F, are on the legs of the angle. Fig. 4.10. Angles are commonly described by their degree measures. The most common angle found in buildings is the right angle, which has a degree measure of exactly 90° and is formed by two lines, rays, or line segments that are perpendicular (⊥) to one another. Some other angles are described by their relation to the right angle...
eBook - ePub- Rajpal S. Sirohi(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...9 Angle Measurement 9.1 DEFINITION OF AN ANGLE The circumference of a circle makes an angle of 360° or 2π rad at its center. If the circumference is divided into 360 equal parts, each segment will make an angle of 1°. The angle is the ratio of circumference to the radius of the circle. Each degree is divided into 60 min of an arc and each minute is divided into 60 s of an arc. Therefore, Angle Measurement is linked to the length and hence is traceable to the standard meter through various kinds of techniques and instrumentation. Angle gauges with different tolerances are used to transfer secondary length standards to the laboratories and industry. We will discuss optical methods to measure angles. These include autocollimator, goniometer, and interferometric methods. 9.2 AUTOCOLLIMATOR Figure 9.1 shows a schematic of an autocollimator, which essentially is a telescope. Filtered radiation illuminates a graticule (a cross-wire), which is placed at the focal plane of a good objective lens. The light exiting from the lens (objective) is thus collimated and falls on a reflecting object. If the object is perpendicular to the optical axis of the autocollimator, it retraces its path and falls on a reticle properly centered, which is also located at the focal plane of the objective. An observer sees the image of the reticle. Alternately, an array detector could be used to sense the position of the image. When the object is not perpendicular to the optical axis but tilted by an angle α, the reflected beam deviates by 2α and the image of the graticule shifts by Δ y (= 2 f α) at the reticle. The reticle is graduated in terms of angles and hence the autocollimator gives direct angle readings. The range of the instrument is limited but it has very high accuracy (~0.1″)...
