Geometric Representation of Angles

 Angles Angles  Initial Side and Standard Position Initial Side and Standard Position

 Degrees: One degree is 1/360 of a revolution.  A right angle is an angle that measures 90 degrees or ¼ revolution  A straight angle is an angle that measures 180 degrees or ½ revolution

 Drawing an Angle  (a) 45 degrees  (b) -90 degrees  (c) 225 degrees  (d) 405 degrees

 1 degree equals 60’ (minutes)  1’ (minute) equals 60” (seconds)  Using graphing calculator to convert

 Definition  Arc Length  For a circle of radius r, a central angle of  radians subtends an arc whose length s is  s=r 

 Find the length of the arc of a circle of radius 2 meters subtended by a central angle of 0.25 radian.  s=r  with r = 2 meters and Θ = 0.25  2(0.25) = 0.25 meter

 One revolution is 2 π therefore, 2 π r = r θ (arc length formula)  It follows then that 2 π = θ and  1 revolution = 2 π radians  360 degrees = 2 π radians  or 180 degrees = π radians  so... 1 degree = π /180 radian and  1 radian = 180/ π degrees

 Convert each angle in degrees to radians:  (a) 60 degrees  (b) 150 degrees  (c) – 45 degrees  (d) 90 degrees

 Convert each angle in radians to degrees  (a) π /6 radian  (b) 3 π /2 radian  (c) -3 π /4  (d) 7 π /3

 Page 375 has common angles in degree and radian measures

 Steps:  (1) Find the measure of the central angle between the two cities  (2) Convert angle to radians  (3) Find the arc length (remember we live on a sphere and the distance between two cities on the same latitude is actually an arc length)

 The area A of the sector of a circle of radius r formed by a central angle of θ radians is  A = ½ r^2 θ  Examples Examples

 Linear Speed:  v = s/t  Angular Speed:  ω = θ /t

 Angular Speed is usually measured in revolutions per minute (rpms).  Converting to radians per minute  Linear Speed given an Angular Speed:  v = r ω  where r is the radius

 A child is spinning a rock at the end of a 2-ft rope at the rate of 180 rpms. Find the linear speed of the rock when it is released.

 At the Cable Car Museum you can see four cable lines that are used to pull cable cars up and down the hills of San Francisco. Each cable travels at a speed of 9.55 miles per hour, caused by rotating wheel whose diameter is 8.5 feet. How fast is the wheel rotating? Express your answer in rpms.

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