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