Angles and Their Measure

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

      It’s Greek To Me! alpha beta gamma theta delta phi It is customary to use small letters in the Greek alphabet to symbolize angle measurement.    alpha beta gamma    theta delta phi

We can use a coordinate system with angles by putting the initial side along the positive x-axis with the vertex at the origin. Quadrant II angle Quadrant I angle Terminal Side  positive  negative Initial Side Quadrant IV angle If the terminal side is along an axis it is called a quadrantal angle. We say the angle lies in whatever quadrant the terminal side lies in.

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

 = - 360° + 45°  = - 315°  = 45°  = 360° + 45° = 405° What is the measure of this angle? You could measure in the positive direction and go around another rotation which would be another 360°  = - 360° + 45°  = - 315°  = 45° You could measure in the positive direction  = 360° + 45° = 405° You could measure in the negative direction There 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.

Acute and Obtuse Angles Acute angles have measure between _____º and _____º. Obtuse angles have measure between ____º and _____º. Straight angles measure _______º.

Classifying Angles Example: 50º is a ____ quadrant angle. Angles are often classified according to the QUADRANT in which their terminal sides lie. Example: 50º is a ____ quadrant angle. 208º is a ____ quadrant angle. II I -75º is a _____ quadrant angle. III IV

Classifying Angles Standard position angles that have their terminal side on one of the axes are called QUADRANTAL ANGLES. For example, 0º, 90º, 180º, 270º, 360º, … are quadrantal angles.

Quadrant III Quadrant I Sketching Angles (Degrees) Sketch in standard position. In which quadrant is  located? Sketch in standard position. In which quadrant is  located? Quadrant III Quadrant I

Complementary and Supplementary Angles Complementary Angles Two positive angles are complementary if their sum is ______º Angles that measure 22º and ____º are complements. Supplementary Angles Two positive angles are supplementary if their sum is _______º Angles that measure 137º and ____º are supplements.

Coterminal angles have the same initial and terminal sides  positive  negative Initial Side +60 and -300 angles are coterminal angles

Find three angles that are coterminal to a 120 angle: 120 - 360 = -240 1 clockwise turn 120 + 360 = 480 1 counterclockwise turn 120 + 2(360) = 840 2 counterclockwise turns

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 minutes 1 minute = 60 seconds  = 25°48'30" degrees seconds minutes To 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 minutes 30" = 0.5'

 = 25°48'30" = 25°48.5' = 25.808° 48.5' = .808° 1 degree = 60 minutes 1 minute = 60 seconds  = 25°48'30" = 25°48.5' = 25.808° Now let's use another conversion fraction to get rid of minutes. 48.5' = .808°

This angle measures 1 radian Another way to measure angles is using what is called radians. Given 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. terminal side arc length is also r r r r initial side This angle measures 1 radian radius of circle is r

Radian Measure Definition of Radian: A 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, for  in radians,

MAT 204 SP09 Radian Measure

Radian Measure

Conversions: Radians Degrees To convert degrees to radians, multiply by To convert radians to degrees, multiply by Converting an angle from to decimal form.

Convert 110o to radians

Conversions Between Degrees and Radians To convert degrees to radians, multiply degrees by To convert radians to degrees, multiply radians by Example Convert from Degrees to Radians: 210º

Conversions Between Degrees and Radians Example a) Convert from radians to degrees: b) Convert from radians to degrees: 3.8

Conversions Between Degrees and Radians Try it! c) Convert from degrees to radians (exact): d) Convert from radians to degrees:

Conversions Between Degrees and Radians Again! e) Convert from degrees to radians (to 3 decimal places): f) Convert from radians to degrees (to nearest tenth): 1 rad

Arc length s of a circle is found with the following formula: IMPORTANT: ANGLE MEASURE MUST BE IN RADIANS TO USE FORMULA! s = r arc length radius measure of angle Find the arc length if we have a circle with a radius of 3 meters and central angle of 0.52 radian. arc length to find is in black  = 0.52 s = r 3 0.52 3 = 1.56 m What 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 2r = r 2 =  2  radians = 360° We need a conversion from degrees to radians. We could use a conversion fraction if we knew how many degrees equaled how many radians. If we look at one revolution around the circle, the arc length would be the circumference. Recall that circumference of a circle is 2r s = r Let's start with the arc length formula 2r = r cancel the r's 2 =  This tells us that the radian measure all the way around is 2. All the way around in degrees is 360°. 2  radians = 360°

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

Area of a Sector of a Circle The formula for the area of a sector of a circle (shown in red here) is derived in your textbook. It is:  r Again  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 Sizes See 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.