Prepared by:Hazelyn C. CarreonMAED-MATH

An angle is formed by joining the endpoints of two half-lines called rays. The side you measure from is called the initial side.Initial SideThe side you measure to is called the terminal side.Terminal SideThis is a counterclockwise rotation.This is a clockwise rotation.Angles measured counterclockwise are given a positive sign and angles measured clockwise are given a negative sign.Positive AngleNegative Angle

Its Greek To Me!It is customary to use small letters in the Greek alphabet to symbolize angle measurement. alphabetagammathetaphidelta

We can use a coordinate system with angles by putting the initial side along the positive x-axis with the vertex at the origin. positiveInitial SideTerminal Side negativeWe say the angle lies in whatever quadrant the terminal side lies in.Quadrant I angleQuadrant II angleQuadrant IV angleIf the terminal side is along an axis it is called a quadrantal angle.

We will be using two different units of measure when talking about angles: Degrees and RadiansLets talk about degrees first. You are probably already somewhat familiar with degrees.If we start with the initial side and go all of the way around in a counterclockwise direction we have 360 degreesYou are probably already familiar with a right angle that measures 1/4 of the way around or 90 = 90If we went 1/4 of the way in a clockwise direction the angle would measure -90 = - 90 = 360

= 45What is the measure of this angle?You could measure in the positive direction = - 360 + 45You could measure in the positive direction and go around another rotation which would be another 360 = 360 + 45 = 405You could measure in the negative directionThere are many ways to express the given angle. Whichever way you express it, it is still a Quadrant I angle since the terminal side is in Quadrant I. = - 315

If the angle is not exactly to the next degree it can be expressed as a decimal (most common in math) or in degrees, minutes and seconds (common in surveying and some navigation).1 degree = 60 minutes1 minute = 60 seconds = 2548'30" degreesminutessecondsTo convert to decimal form use conversion fractions. These are fractions where the numerator = denominator but two different units. Put unit on top you want to convert to and put unit on bottom you want to get rid of.Let's convert the seconds to minutes30" = 0.5'

1 degree = 60 minutes1 minute = 60 seconds = 2548'30" Now let's use another conversion fraction to get rid of minutes.48.5' = .808 = 2548.5' = 25.808

initial sideterminal sideradius of circle is rrrarc length is also rrThis angle measures 1 radianGiven a circle of radius r with the vertex of an angle as the center of the circle, if the arc length formed by intercepting the circle with the sides of the angle is the same length as the radius r, the angle measures one radian.Another way to measure angles is using what is called radians.

Arc length s of a circle is found with the following formula:arc lengthradiusmeasure of angleIMPORTANT: ANGLE MEASURE MUST BE IN RADIANS TO USE FORMULA!Find the arc length if we have a circle with a radius of 3 meters and central angle of 0.52 radian.3 = 0.52arc length to find is in blacks = r= 1.56 mWhat if we have the measure of the angle in degrees? We can't use the formula until we convert to radians, but how?s = r

We need a conversion from degrees to radians. We could use a conversion fraction if we knew how many degrees equaled how many radians.Let's start with the arc length formulas = rIf we look at one revolution around the circle, the arc length would be the circumference. Recall that circumference of a circle is 2r2r = rcancel the r'sThis tells us that the radian measure all the way around is 2. All the way around in degrees is 360.2 = 2 radians = 360

2 radians = 360Convert 30 to radians using a conversion fraction.30The fraction can be reduced by 2. This would be a simpler conversion fraction.180 radians = 180Can leave with or use button on your calculator for decimal.Convert /3 radians to degrees using a conversion fraction. 0.52= 60

Area of a Sector of a CircleThe formula for the area of a sector of a circle (shown in red here) is derived in your textbook. It is:rAgain must be in RADIANS so if it is in degrees you must convert to radians to use the formula.Find the area of the sector if the radius is 3 feet and = 50= 0.873 radians

A Sense of Angle SizesSee if you can guess the size of these angles first in degrees and then in radians.You will be working so much with these angles, you should know them in both degrees and radians.

*Measuring AnglesThe measure of an angle is determined by the amount of rotation from the initial side to the terminal side.

There are two common ways to measure angles, in degrees and in radians.Well start with degrees, denoted by the symbol .One degree (1) is equivalent to a rotation of of one revolution.

Measuring Angles

Section 4.1, Figure 4.13, Common Degree Measures on the Unit Circle, pg. 251

Angles are often classified according to the quadrant in which their terminal sides lie.

Ex1: Name the quadrant in which each angle lies.50208 II I-75 III IV

Classifying AnglesQuadrant 1Quadrant 3Quadrant 4

Classifying AnglesStandard position angles that have their terminal side on one of the axes are called quadrantal angles. For example, 0, 90, 180, 270, 360, are quadrantal angles.

Coterminal AnglesAngles that have the same initial and terminal sides are coterminal.

Angles and are coterminal.

Section 4.1, Figure 4.4, Coterminal Angles, pg. 248

Example of Finding Coterminal AnglesYou can find an angle that is coterminal to a given angle by adding or subtracting multiples of 360.

Ex 2: Find one positive and one negative angle that are coterminal to 112.For a positive coterminal angle, add 360 : 112 + 360 = 472 For a negative coterminal angle, subtract 360: 112 - 360 = -248

Ex 3. Find one positive and one negative angle that is coterminal with the angle = 30 in standard position.

Ex 4. Find one positive and one negative angle that is coterminal with the angle = 272 in standard position.

Radian MeasureA second way to measure angles is in radians.

Definition of Radian:One radian is the measure of a central angle that intercepts arc s equal in length to the radius r of the circle.

In general,

Section 4.1, Figure 4.5, Illustration of Arc Length, pg. 249

Radian Measure

Section 4.1, Figure 4.6, Illustration of Six Radian Lengths, pg. 249

Radian Measure

Section 4.1, Figure 4.7, Common Radian Angles, pg. 249

*6.1 Radian and Degree MeasureConversions Between Degrees and RadiansTo convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by

Ex 5. Convert the degrees to radian measure.

60

30

-54

-118

45

Ex 6. Convert the radians to degrees.a)

b)

c)

d)

Ex 7. Find one positive and one negative angle that is coterminal with the angle = in standard position.

Ex 8. Find one positive and one negative angle that is coterminal with the angle = in standard position.

*

Class WorkConvert from degrees to radians.54-300

Convert from radians to degrees.3.

4.

Find one postive angle and one negative angle in standard position that are coterminal with the given angle.135

References:Active Math (Advanced Algebra, Trigonometry & Statistics) by Jude Ildefonso P. Tidalgo and Mariano M. Melendres, Innovative Educational Materials, Inc., 2011, pp. 195-208Advanced Algebra, Trigonometry & Statistics: Patterns and Practicalities by Minie Rose C. Lapinid and Olivia N. Buzon, Don Bosco Press, Inc., 2007, pp.195-209http://www.themathpage.com/aTrig/measure-angles.htmhttps://www.youtube.com/watch?v=0-hgeXamzEohttp://www.regentsprep.org/regents/math/algtrig/ATT3/referenceAngles.htm