Geometric Representation of Angles

Angles

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=rwith r = 2 meters and Θ = 0.25

2(0.25) = 0.25 meter

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

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) 3π/2 radian (c) -3π/4 (d) 7π/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^2θ

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 = rω 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.

On-line Examples

On-line Tutorial