ANGLES AND THEIR MEASURE Geometric Representation of Angles

ANGLES AND THEIR MEASURE Geometric Representation of Angles

Definition of Angles � Initial Side and Standard Position

Angles � 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

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

Converting between Degrees, Minutes, Seconds and Decimal � 1 degree equals 60’ (minutes) � 1’ (minute) equals 60” (seconds) � Using graphing calculator to convert

Radians � Definition � Arc Length For a circle of radius r, a central angle of q radians subtends an arc whose length s is � � s=r q

Finding the Length of a Circle � Find the length of the arc of a circle of radius 2 meters subtended by a central angle of 0. 25 radian. s=r q with r = 2 meters and Θ = 0. 25 � 2(0. 25) = 0. 25 meter �

Relationship Between Degrees and Radians � � � � One revolution is 2π therefore, 2πr = rθ (arc length formula) It follows then that 2π = θ and 1 revolution = 2π radians 360 degrees = 2π radians or 180 degrees = π radians so. . . 1 degree = π/180 radian and 1 radian = 180/π degrees

Converting from Degrees to Radians � Convert each angle in degrees to radians: � (a) 60 degrees (b) 150 degrees (c) – 45 degrees (d) 90 degrees � � �

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

Common Angles in Degrees and Radians � Page 375 has common angles in degree and radian measures

Finding Distance Between two Cities � � 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)

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

Circular Motion � Linear Speed: � v = s/t � Angular Speed: � ω = θ/t

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

Finding Linear Speed � 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.

Cable Cars of San Francisco � 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.

Circular Motion � On-line Examples � On-line Tutorial