PreCalculus 4/5/13 Obj: SWBAT use degree and radian measure

Transcription

1 PreCalculus 4/5/13 Obj: SWBAT use degree and radian measure Agenda Go over DMS worksheet Go over last night 1-31 #3,7,13,15 (put in bin) Complete 3 slides from Power Point 11:00 Quiz 10 minutes (grade it!) Time permits Go over Chapter 3 Test Homework: p360 #33-41 odds, 72 Announcements: 30 th Week Exam 4/10

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3 Ex Concentric Circles circle within a circle The concentric circles on an archery target are 6 inches apart. The inner circle (light blue aqua) has a perimeter of 37.7 inches. What is the perimeter of the next-largest (red) circle? What is the area of the red circle Circumference = Perimeter = 2πr Ans in Ans. 108π sq.in sq. in.

4 Ex A 100 degree sector cut from a circular disc has a length of 7 cm. To the nearest cm., what is the radius of the circle? What is the area of the sector? S = r ɵ ɵ Must be in radians Just convert degrees to radians Ans. 4 cm Ans sq. cm

5 Ex In navigation the Course or Bearing of an object is the angle of the line of travel measured clockwise from due north. In the figure, the line of travel has the bearing of 155 degrees.

6 6.1 Radian and Degree Measure 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 6

7 4.1 Radian and Degree Measure Angles Trigonometry: measurement of triangles 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

8 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. Digital Figures, 4 3 The initial side of an angle in standard position is always located on the positive x-axis. 8

9 Measuring Angles The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. Section 4.1, Figure 4.1, Terminal and Initial Side of an Angle, pg. 248 There are two common ways to measure angles, in degrees and in radians. Copyright Houghton Mifflin Company. All rights reserved. Digital Figures, 4 2 ****************************************************************** We ll start with degrees, denoted by the symbol º. 9

10 One degree (1º) is equivalent to a rotation of of one revolution.

11 Measuring Angles 4.1 Radian and Degree Measure Section 4.1, Figure 4.13, Common Degree Measures on the Unit Circle, pg Copyright Houghton Mifflin Company. All rights reserved. Digital Figures,

12 Radian Measure 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. IMPORTANT: ANGLE MEASURE MUST BE IN RADIANS TO USE FORMULA! In general, for in radians, S = arc length r = radius s r = central angle Central angle has vertex at center of Circle 12

13 Section 4.1, Figure 4.7, Common Radian Angles, pg. 249 Radian Measure Section 4.1, Figure 4.13, Common Degr Measures on the Unit Circle, pg. 251 Copyright Houghton Mifflin Company. All rights reserved. Digital Figures, 4 8 Degree Measure 13

14 Arc length s of a circle is found with the following formula: s = r arc length radius measure of angle IMPORTANT: 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 = 0.52 arc length to find is in black s = 3 r 0.52 = 1.56 m

15 3 min. Class Work Ans. 70 cm Ans cm What if we have the measure of the angle in degrees? We can't use the formula until we convert to radians, but how?

16 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 formula cancel the r's s = r 2 r = r 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 2 = 2 radians = 360 This tells us that the radian measure all the way around is 2. All the way around in degrees is 360.

17 Conversions Between Degrees and Radians 1. To convert degrees to radians, multiply degrees by 2. To convert radians to degrees, multiply radians by Example Convert from Degrees to Radians: 210º

18 Conversions Between Degrees and Radians Example a) Convert from radians to degrees: Ans degrees b) Convert from radians to degrees: Ans. 684/π = degrees 18

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

20 Conversions Between Degrees and Radians Again! e) Convert from degrees to radians (to 3 decimal places): 52 Ans..908 radians 52 f) Convert from radians to degrees (to nearest tenth): 1 rad 1 Ans degrees 20