MATHIvProjectANGLES AND THEIR MEASUREMENT

Novie I. Fauni Mary Grace RovedilloIV- Plato Math IV Teacher

I. INTRODUCTIONTrigonometry(fromGreektrignon, "triangle" andmetron, "measure"[1]) is a branch ofmathematicsthat studies relationships involving lengths andanglesoftriangles. The field emerged during the 3rd century BC from applications ofgeometryto astronomical studies.[2]The 3rd-century astronomers first noted that the lengths of the sides of a right-angle triangle and theanglesbetween those sides have fixed relationships: that is, if at least the length of one side and the value of one angle is known, then all other angles and lengths can be determined algorithmically. These calculations soon came to be defined as thetrigonometric functionsand today are pervasive in bothpureandappliedmathematics: fundamental methods of analysis such as theFourier transform, for example, or thewave equation, use trigonometric functions to understandcyclicalphenomena across many applications in fields as diverse as physics, mechanical and electrical engineering, music and acoustics, astronomy, ecology, and biology. Trigonometry is also the foundation ofsurveying.Trigonometry is most simply associated withplanarright-angletriangles (each of which is a two-dimensional triangle with one angle equal to 90 degrees). The applicability to non-right-angle triangles exists, but, since any non-right-angle triangle (on a flat plane) can be bisected to create two right-angle triangles, most problems can be reduced to calculations on right-angle triangles. Thus the majority of applications relate to right-angle triangles. One exception to this isspherical trigonometry, the study of triangles onspheres, surfaces of constant positivecurvature, inelliptic geometry(a fundamental part ofastronomyandnavigation). Trigonometry on surfaces of negative curvature is part ofhyperbolic geometry.Trigonometry is the study of angles and relationships between them. Especially important in trigonometry are the angles of a triangle. For this reason, trigonometry is closely linked with geometry. One of the major differences between trigonometry and geometry, though, is that trigonometry concerns itself with actual measurements of angles and sides of a triangle, whereas geometry focuses on establishing relationships between unmeasured angles and sides. To begin our study of trigonometry, we'll review the definition and some characteristics of angles to make sure we have a solid foundation for learning more about them.Angles, by definition, lie in a plane, so trigonometry is a two-dimensional field of study. It will be convenient, and eventually necessary, to become familiar with the coordinate plane, which is a system of measuring and plotting points in two dimensions. The location of any point in a plane, then, can be specified by exact coordinates. A point can also be specified by a vector. A vector is like a line segment lying in a specific position--it has length and direction. Vectors can be used to determine the location of points, as well as the measure of certain angles. These basic concepts will provide a foundation for understanding the principles of trigonometry.

II. DISCUSSIONTRIGONOMETRY, as it is actually used in calculus and science, is not about solving triangles. It becomes the mathematical description of things that rotate or vibrate, such as light, sound, the paths of planets about the sun, or satellites about the earth. It is necessary therefore to have angles of any size, and to extend to them the meanings of the trigonometric functions.AnglesAnangleis the opening that two straight lines form when they meet.

When the straight line FA meets the straight line EA, they form the angle we name as angle FAE. Letter A, which we place in the middle, labels the point where the two lines meet, and is called thevertexof the angle. When there is no confusion as to which point is the vertex, we may speak of "the angle at the point A," or simply "angle A."The two straight lines that form an angle are called itssides. Andthe sizeof the angle does not depend on the lengths of its sides. We can see that in the figure above. For if the point C is in the same straight line as FA, and B is in the same straight line as EA, then angles CAB and FAE are the same angle.Now, to measure an angle, we place the vertex at the center of a

circle (we call that acentral angle), and we measure the length of thearc-- that portion of the circumference -- that the sides intercept. We then determine what relationship that arc has to the entire circumference, which is an agreed-upon number. (In degree measure that number is 360; in radian measure it is 2.)The measure of angle A, then, will be length of the arc BC relative to the circumference BCD -- or the length of arc EF relative to the circumference EFG. Forin any circles, equal central angles determine a unique ratio of arc to circumference. (See thetheoremof Topic 14. It is stated there in terms of the ratio of arc to radius, but the circumference is proportional to the radius: C = 2r.)There are two systems for measuring angles. One is the well-known system of degree measure. The other is the strictly mathematical system called radian measure.

An angle is the union of two rays that share a common endpoint. The rays are called the sides of the angle, and the common endpoint is the vertex of the angle. The measure of an angle is the measure of the space between the rays. It is the direction of the rays relative to one another that determine the measure of an angle.In trigonometry, angles are often defined in terms of rotation. Consider one ray, and then let it rotate a fixed distance about its endpoint. The ray in its initial position before the rotation, and the ray in its ending, or terminal position, after the rotation, creates an angle. The endpoint point about which the ray rotates is the vertex. The amount of rotation determines the measure of the angle. The ray in the initial position, before the rotation, is called the initial side of the angle. The ray in the terminal position, after the rotation, is called the terminal side of the angle. An angle created this way has a positive measure if the rotation was counterclockwise, and a negative measure if the rotation was clockwise.

MEASURING ANGLESThere are three units of measure for angles: revolutions, degrees, and radians. In trigonometry, radians are used most often, but it is important to be able to convert between any of the three units.RevolutionsA revolution is the measure of an angle formed when the initial side rotates all the way around its vertex until it reaches its initial position. Thus, the terminal side is in the same exact position as the initial side. In trigonometry, angles can have a measure of many revolutions--there is no limit to the magnitude of a given angle. A revolution can be abbreviated "rev".Degree measureTo measure an angle in degrees, we imagine the circumference of a circle divided into 360 equal parts, and we call each of those equal parts a "degree." Its symbol is a small 0: 1 -- "1 degree." The full circle, then, will be 360. But why the number 360? What is so special about it? Why not 100 or 1000?The answer is two-fold. First, 360 has many divisors, and therefore it will have many whole number parts. It has an exact half and an exact third -- which a power of 10 does not have. 360 has a fourth part, a fifth, a sixth, and so on. Those are natural divisions of the circle, and it is very convenient for their measures to be whole numbers. (Even the ancients didn't like fractions)Secondly, 360 is close to the number of days in the astronomical year: 365.

The measure of an angle, then, will be as many degrees as its sides include. To say that angle BAC is 30 means that its sides enclose 30of those equal divisions. Arc BC is30360of the entire circumference.

So, when 360 is the measure of a full circle, then 180 will be half a circle. 90 -- one right angle -- will be a quarter of a circle; and 270 will be three quarters of a circle: three right angles.

A more common way to measure angles is in degrees. There are 360 degrees in one revolution. Degrees can be subdivided, too. One degree is equal to 60 minutes, and one minute is equal to 60 seconds. Therefore, an angle whose measure is one second has a measure ofdegrees. When perpendicularity is discussed, it is most often defined as a situation in which a 90 degree angle exists. Often degrees are used to describe certain triangles, like30-60-90and45-45-90triangles. As previously mentioned, however, in most cases that concern trigonometry, radians are the most useful and manageable unit of measure. Degrees are symbolized with a small superscript circle after the number (measure). 360 degrees is symbolized360o.RadianA radian is not a unit of measure that is arbitrarily defined, like a degree. Its definition is geometrical. One radian (1 rad) is the measure of the central angle (an angle whose vertex is the center of a circle) that intercepts an arc whose length is equal to the radius of the circle. The measure of such an angle is always the same, regardless of the radius of the circle. It is a naturally occurring unit of measure, just likeis the natural ratio of the circumference of a circle and the diameter. If an angle of one radian intercepts an arc of lengthr, then a central angle of2radians would intercept an arc of length2r, which is the circumference of the circle. Such a central angle has a measure of one revolution. Therefore,1rev = 360o= 2rad. Also,1rad = ()o=rev.Conversion between Revolutions, Degrees, and RadiansBelow is a chart with angle measures of common angles in revolutions, degrees, and radians. Any angle can be converted from one set of units to another using the definition of the units, but it will save time to memorize a few simple conversions. It is particularly important to be able to convert between degrees and radians.

The Coordinate PlaneAngles lie in a plane. To specify the point in space where an angle lies, or where any figure exists, a plane can be assignedcoordinates.Since a plane is two-dimensional, only two coordinates are required to designate a specific location for every point in the plane. One coordinate determines the length, and the other determines width. In reality, length and width are the same thing--they are used because they describe distance in two directions which are perpendicular to each other. This is all the coordinate plane is: a plane with two perpendicular axes by which distance in either of two dimensions can be measured.The coordinate plane consists of an origin and two axes. The origin is a point. The axes are lines perpendicular to each other that intersect at the origin. Below is pictured the coordinate plane, with the origin at point O.

The origin is fixed, and designated as the point (0,0). Every other point is assigned an ordered pair,(x,y), according to its position relative to the origin. The two axes are named the $x$-axis and the y-axis. In most drawings, thex-axis is the horizontal axis, and the y-axis is the vertical axis, but this does not necessarily need to be the case. A point is assigned an ordered pair consisting of two real numbers: The first is the x-coordinate, which measures how far the point is from the y-axis. The second real number making up an ordered pair is the y-coordinate, which measures the distance between the point and the x-axis. Often the axes are pictured with tick marks indicating length to make it easier to measure distance. When a point is drawn into the coordinate plane and assigned an ordered pair, it is plotted. Take a gander at the plotted points below.

Note that some of the coordinates are negative numbers. Negative distance does not exist, but coordinates are given either positive or negative values to specify which side of the given axis they are on. In most cases, the positive direction of the x-axis points to the right, and the positive direction of the y-axis points upward. Thus, for example, points on the left of the y-axis have a negative x-coordinate. The positive directions don't always have to be these directions, though. Often, as in the diagram above, the axes will only have an arrow on the end which points in the positive direction. The other end has no arrow. This is how one can tell where the positive and negative values lie.A plane extends in all direction without limit. So does the coordinate plane. Although there are many ways to draw the coordinate plane, it is always the same thing: a point of origin and two axes, which intersect at the origin and lie perpendicular to each other. The origin, by definition, always has the coordinates (0,0). Every other point in the plane can be measured according to the axes. Even the point (33563452143,23455434) exists and can be located in any coordinate plane; it extends without limit. Below are some other ways to draw the coordinate plane. All look different, but they are all the same coordinate plane.

The axes of the coordinate plane divide the plane into four regions--these regions are called quadrants. The region in which the x-coordinate and the y-coordinate are both positive is called Quadrant I. Quadrant II is the region in whichx< 0andy> 0. Quadrant III is the region in whichx< 0andy< 0. Quadrant IV is the region in whichx> 0andy< 0. The quadrants are labeled in the figure below.

VectorsOne way to represent motion between points in the coordinate plane is with vectors. A vector is essentially a line segment in a specific position, with both length and direction, designated by an arrow on its end. The figures below are vectors.

A vector can be named by a single letter, such asv. The vectorvis symbolized by a lettervwith an arrow above it, like this:. A vector is determined by two coordinates, just like a point--one for its magnitude in thexdirection, and one for its magnitude in theydirection. The magnitude of a vector in the x-direction is called the horizontal, or x-component of the vector. The magnitude of a vector in the y-direction is called the vertical, or $y$-component of the vector. A vectorwith coordinates (3,4) and origin at the origin of the coordinate plane looks like this:

A vector has length and direction, that is all. Two vectors with the same length and direction are the same vector. They may have origins at different points, but they are still equal. The length of a vector is formally called its magnitude. Given the coordinates of a vector(x,y), its magnitude is. This formula is drawn from the **Pythagorean Theorem* {math/geometry2/specialtriangles}*. The direction of a vector is only fixed when that vector is viewed in the coordinate plane. Then, using techniques we'll learn shortly, the direction of a vector can be calculated. Outside the coordinate plane, directions only exist relative to one another, so a single vector cannot have a specific direction.

Operations with VectorsVectors can be added and subtracted to one another, and multiplied and divided by scalars (number with magnitude but no direction). When two vectors are added or subtracted, the x-component of one vector is added or subtracted to the x-component of the other, and the same is done with the y-components of the vectors. For example, ifand, then. When a vector is multiplied or divided by a scalar, the scalar (any real number) is simply distributed through to both coordinates of the vector. Hence, using the vectors defined above,2and. In any case, the sum, difference, product, or quotient is still a vector.A vector whose origin is the origin of the coordinate plane ends at the point with the same coordinates as the vector. Because vectors have a fixed magnitude, they always determine two points, the origin of the vector and the endpoint. Vectors are useful mathematical tools for modeling motion and symbolizing directed line segments.Vectors vs. RaysOne more note is important to make in this lesson: vectors are not rays. They are symbolized the same way--a line segment with an arrow on one end--but they are very different things. Vectors have a specified length, rays have infinite length. From this point on, whenever a line semgent is drawn with an arrow on one end, assume that it is a ray. If such a figure is a vector, it will be noted.Standard position

We say that an angle is instandard positionwhen its vertex A is at theoriginof the cordinate system, and itsInitial sideAB lies along the positivex-axis. We say that AB has "swept out" the angle BAC, and that AC is itsTerminal side.

We now think of the terminal side AC as rotating about the fixed point A. When it rotates in a counter-clockwise direction, we say that the angle ispositive. But when it rotates in a clockwise direction, as AC', the angle isnegative.When the terminal side AC has rotated 360, it has completed one full revolution.Angles can exist anywhere in the coordinate plane where two rays share a common vertex. If this vertex is at the origin of the plane and the initial side lies along the positive $x$-axis, then the angle is said to be in standard position. Some angles in standard position are shown below.

Angles in standard position can be classified according to the quadrant contains their terminal sides. For example, an angle whose terminal side lies in the first quadrant is called a first quadrant angle. If the terminal side of an angle lies along one of the axes, then that angle doesn't lie in one specific quadrant; it lies along the border of two quadrants. Such angles are called a quadrantal angle.The four quadrants

Thex-yplane is divided into fourquadrants. The angle begins in itsstandard positionin the first quadrant ( I ). As the angle continues -- in the counter-clockwise direction -- we name each succeeding quadrant.Why do we name the quadrants in the counter clockwise direction? Because in what we call the "first" quadrant, the algebraic signs ofxandyare positive.Coterminal anglesAngles arecoterminalif, when in thestandard position, they have the same terminal side.

For example, 30 is coterminal with 360 + 30 = 390. They have the same terminal side. That is, their terminal sides are indistinguishable.Any angleis coterminal with+ 360 -- because we are just going around the circle one complete time.90 is coterminal with 270. Again, they have the same terminal side.Notice: 90plus270 = 360. The sum of the absolute values of those coterminal angles completes the circle.

III. QUIZProblem 1.How many degrees corresponds to each of the following?a) A third of a revolution b) A sixth of a revolution c) Five sixths of a revolution d) Two revolutions e) Three revolutions f) One and a half revolutions Example 1.30 is what fraction of a circle, or of one revolution?Answer.30 is30360of a revolution:

IV. 30360=336=112

Problem 2.What fraction of a revolution is each of the following? a) 60

b) 45

c) 72

Example 2.If the diameter of a circle is 16 cm, how long is the arc intercepted by acentral angleof 45?

Answer.45 is one eighth of a full circle. (It is half of 90 , which is one quarter.) Now, the full circumference of this circle isC =D = 3.14 16 cm. The intercepted arc is one eighth of the circumference:3.14 16 8 = 3.14 2 = 6.28 cm

Problem 3.If the diameter of a circle is 20 in, how long is the arc intercepted by a central angle of 72?

Problem 4.In which quadrant does each angle terminate? a) 15 b) 15 c) 135 d) 390 e) 100 f) 460

Problem 5.Name the non-negative angle that is coterminal with each of these, and is less than 360. a) 360 b) 450 c) 20 d) 180 e) 270 f) 720 g) 200

KEY ANSWERSProblem 1.A) A third of 360 = 360 3 = 120B) 360 6 = 60C) 5 60 = 300D) 2 360 = 720E) 3 360 = 1080F) 360 + 180 = 540

Problem 2.A)60 = 6 = 1/636 360B)45/360 =5/40 = 1/8

C)72/360 = 8/40 = 1/5

Problem 3.We saw in Problem 2c) that 72 is one fifth of a circle. The circumference of this circle is C =D = 3.14 20 in. The intercepted arc is one fifth of this: 3.14 20 5 = 3.14 4 = 12.56in.

Problem 4.a) I b) IV c) II d) I. 390 = 360 + 30 e) III f) III. 460 = 360 100g) IV.710 is 10lessthan two revolutions, which are 720.

Problem 5.a)0b) 90. 450 = 360 + 90c)340 d) +180e) 90f) 0.720 = 2 360g)160

REFERENCEhttp://www.sparknotes.com/math/trigonometry/angles/http://www.themathpage.com/atrig/measure-angles.htm

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