Angles and Their Measurements Prepared By: Still John F. Reyes 1 Rectangular Coordinate System Definitions • • • • • • • X-axis – the horizontal axis Y –axis – the vertical axis Rectangular coordinate system: where the xaxis and the y-axis intersect Ordered pair: identifies the location of a point on the rectangular coordinate system (x coordinate, y coordinate) Origin: (0,0) – intersection of the x and y axis Quadrant: a section of the rectangular coordinate system 2 3 How to graph a point? First Locate the origin Second, graph your x -coordinate: IF the value of your x coordinate is positive move to the right, if negative move to the left Third, graph your y coordinate: IF the value of your ycoordinate is positive move up, if negative move down How to identify the quadrant? First – locate where your point is Second – check your graph on which quadrant it is in. In this section, we will study the following topics: Terminology used to describe angles Degree measure of an angle Radian measure of an angle Converting between radian and degree measure Coterminal angles Angles in a triangle 6 Angles Section 4.1, Figure 4.1, Terminal and Initial Side of an Angle , pg. 248 Angle Measure Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–2 7 Standard Position Section 4.1, Figure 4.2, Standard Position of an Angle, pg. 248 Vertex at origin Copyright © Houghton Mifflin Company. All rights reserved. The initial side of an angle in standard position is always located on the positive x-axis. Digital Figures, 4–3 8 Positive and negative 4.3, Positive and 4.1, Figureangles Section Negative Angles, pg. 248 When sketching angles, always use an arrow to show direction. Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–4 9 Measuring Angles The 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. We’ll start with degrees, denoted by the symbol º. 1 360 One degree (1º) is equivalent to a rotation of of one revolution. 10 Section 4.1, Figure 4.13, Common Degree Measures on the Unit Circle, pg. 251 Measuring Angles 1 360 Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–9 11 Classifying Angles Angles are often classified according to the quadrant in which their terminal sides lie. Ex: Name the quadrant in which each angle lies. 50º Quadrant 1 208º Quadrant 3 II I -75º Quadrant 4 III IV 12 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. 13 Coterminal Angles AnglesSection that have theFigure same 4.4, initial and terminal 4.1, Coterminal Angles, pg. 248 sides are coterminal. Angles and are coterminal. Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–5 14 Example of Finding Coterminal Angles You can find an angle that is coterminal to a given angle by adding or subtracting multiples of 360º. Ex: Find one positive and one negative angle that are coterminal to 112º. For a positive coterminal angle, add 360º : 112º + 360º = 472º For a negative coterminal angle, subtract 360º: 112º - 360º = -248º 15 Degree System of Angular Measure 1° (one degree) = 60’ (60 minutes) 1’ (one minute) = 60” (60 seconds) 360° (360 degrees) = 1 revolution There are two ways of measuring angles: 1. Degrees, Minutes, Seconds (DMS) 2. Decimal Degrees Examples: Express each angle measure using DMS 1) 10.5° 2) 42.72° Express the ff. in decimal degrees 1) 95° 45’ 45” Exercises 1. 2. 3. 4. Find one positive and one negative angle that is coterminal with the angle = 30° in standard position. Find one positive and one negative angle that is coterminal with the angle = 272 in standard position. Express 27.3° using DMS. Express 28° 18’ 36” in decimal degrees. 18 Do Worksheet 1 19 Radian Measure A second way to measure angles is in radians. Definition of Radian: One radian is the4.1, measure a central angleof that intercepts Section Figureof4.5, Illustration arc s equal in length to the radius r of the circle. Arc Length, pg. 249 In general, s r Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–6 20 Radian Measure 2 radians corresponds to 360 2 6.28 3.14 radians corresponds to 180 Section 4.1, radians corresponds toFigure 90 4.6, Illustration of 1.57 Six Radian Lengths, pg. 249 2 2 Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–7 21 Section 4.1, Figure 4.7, Common Radian Measure Radian Angles, pg. 249 22 Conversions Between Degrees and Radians 1. To convert degrees to radians, multiply degrees by 180 2. To convert radians to degrees, multiply radians by 180 23 Ex. Convert the degrees to radian measure. a) 60 b) 30 Ex. Convert the radians to degrees. a) rad 6 b) rad 2 Ex. Find one positive and one negative angle that is coterminal with the angle = in standard position. 3 Ex. Find one positive and one negative angle that is 7 coterminal with the angle = in standard position. 5 Interior Angles of a Triangle The sum of the interior angles of any triangle is 180°. In a right triangle, the sum of the acute angles is 90°. The two angles are said to be complementary angles. Two angles with a sum of 180° is said to be supplementary angles. 27 Examples 1) 2) Given: ∆ABC, if angle A=95°, angle B=45°, find angle C. Given: ∆ABC is a right triangle and angle B=46° 20’, find angle A. 28 Exercises 29 Applications of Angle Measures 30 Arc Length and Central Angles s r s arc length (length of the intercepte d arc) central angle measured in radians r radius Example Find the measure of a rotation in radians when a point 2 m from the center of rotation travels 4 m. s r 4 2 2 Example Find the length of an arc of a circle of radius 5 cm associated with an angle of /3 radians. s r s 5 3 5 s 3 Linear & Angular Velocity Things that turn have both a linear velocity and an angular velocity. Linear Velocity Definition: Linear Velocity is distance/time: Linear Speed s v t Distance Time Angular Velocity Definition: Angular Velocity is turn/time: Angular Speed (omega) t Rotation in radians Time Linear & Angular Velocity Definition of Linear Velocity: Recall Arc Length Formula Linear Velocity in terms of Angular Velocity: s v t r v t v r Let us take 2 pendulums hung on a slim rotating rod for analysis. If the 2 pendulums (A and B) rotate one full cycle, the time taken by them is the same. They covered the same amount of angular distance (360 degree) within the same amount of time. This showed that they have exactly the SAME angular speed. But is the Linear speed the same? Let us take 2 pendulums hung on a slim rotating rod for analysis. The length of the 2 circumferences travelled by the individual pendulums are not the same. The linear length or distance is therefore NOT the same. Length = 2 x (pi) x radius = 2πr They took the same time to complete one full cycle, though. The linear speed is thus DIFFERENT, having travelled different length for the same amount of time. Example: A satellite traveling in a circular orbit approximately 1800 km. above the surface of Earth takes 2.5 hrs. to make an orbit. The radius of the earth is approximately 6400 km. a) Approximate the linear speed of the satellite in kilometers per hour. b) Approximate the distance the satellite travels in 3.5 hrs. 1800 km 6400 km Earth Not to scale Example: r = 6400 + 1800 = 8200 t = 2.5 hrs. a) Approximate the linear speed of the satellite in kilometers per hour. r v t 82002 v 2 .5 v 6560 km / hr 1800 km 6400 km Earth Not to scale Example: r = 6400 + 1800 = 8200 v 6560 km / hr t = 2.5 hrs. b) Approximate the distance the satellite travels in 3.5 hrs. s v t s 6560 3.5 s 22960 km 1800 km 6400 km Earth Not to scale A small pulley 6 cm in diameter is connected by a belt to a larger pulley 15cm in diameter. The small pulley is turning at 120 revolutions/rotations per minute (rpm). a) b) Find the angular velocity of the small pulley in radians per second. Find the linear velocity of the rim of the small pulley. A small pulley 6 cm in diameter is connected by a belt to a larger pulley 15cm in diameter. The small pulley is turning at 120 rpm. a) Find the angular velocity of the small pulley in radians per second. 120 rotations 2 1 min 1 min 1rotation 60 sec 4 radians / sec A small pulley 6 cm in diameter is connected by a belt to a larger pulley 15cm in diameter. The small pulley is turning at 120 rpm. 4 radians / sec b) Find the linear velocity of the rim of the small v r pulley. v 3cm4 radians / sec v 12 cm / sec Exercises 1. 2. 3. Find the length of the arc intercepted by a central angle of measure 2.5 radians in a circle whose radius is 15 cm. A central angle of 38° intercepts an arc of 5 meters. Find the radius of the circle. A flywheel 6 ft. in diameter makes 40 rpm. Find its angular velocity in radians per second. b. Find the speed of the belt that drives the flywheel. a. 46 Do Worksheet 2 47