7.1 Angles and their Measures
It can be shown that linear speed is dependent on the radius of the circle and how fast the object is rotating:
(v= s/t= r⍬/t= r* ⍬/t= r*⍵) v=r*⍵ where ⍵ is measured in radians per unit time, t
conversion proportion formula
(degree/180°)= (radian/π)
2π/3= ?°
120°
Area of a Sector equation
A= 1/2r^2⍬
If radius and arc length same value, measure of ⍬ is ? radian
1
3π/4= ?°
135°
5π/6= ?°
150°
π= ?°
180°
7π/6= ?°
210°
5π/4= ?°
225°
4π/3= ?°
240°
3π/2= ?°
270°
5π/3= ?°
300°
π/6= ?°
30°
7π/4= ?°
315°
11π/6= ?°
330°
2π= ?°
360°
one revolution measurements (degrees and radians)
360°= 2π radians
π/4= ?°
45°
π/3= ?°
60°
π/2= ?°
90°
quadrantal angle
an angle in standard position whose terminal side coincides with one of the axes
If two rays are drawn with a common vertex, they form an ? One ray of an angle is called the ? side and the other ? side. The angle formed is identified by showing the direction and amount of rotation from the initial side to the terminal side. If the the rotation is in the ? direction, the angle is positive; if the rotation is ?, the angle is negative.
angle; initial, terminal; counterclockwise, clockwise
the ? ⍵ of an object is the angle ⍬ (measured in radians) swept out, divided by the elapsed time t, ⍵= ⍬/t (angle in radian/time)
angular speed
Arc Length Theorem
for a circle of radius r, a central angle ⍬ radians subtends an arc whose length s is s=r⍬
Suppose that an object moves around a circle of radius r at a constant speed. If s is the distance traveled in time t around this circle, then the ? v of the object is defined as v=s/t (distance/time)
linear speed
In summary, ? measures how fast the position of an object is changing and ? measures how fast and angle is changing. An object traveling in a circular motion has both linear and angular speed.
linear speed, angular speed
Conversion formula for degrees to radians
rad=⍬(π/180)
Angles are measured by determining the amount of rotation needed for the initial side to become coincident with the terminal side. The two commonly used measures for angles are ? and ?
radians, degrees
A ? is a portion of a line that starts at point v on the line and extends indefinitely in one direction. The starting point v of a ray is called its ?
ray; vertex
Arc Length Theorem equation
s= r⍬
An angle ⍬ is said to be in ? if its vertex at the origin of a rectangular coordinate system and its initial side coincides with the positive x-axis
standard position
Area of a Sector theorem
the area A of the sector of a circle of radius r formed by a central angle of ⍬ radians is 1/2r^2⍬
Equation for Linear Speed
v=s/t
180°= ? radians
π radians
Conversion formula for radians to degrees
⍬=180(rad/π)
Equation for angular speed
⍵= ⍬/t