6.1: Angle Measure in degrees

How to measure angles

Numbers on protractor = angle measure in degrees

I 1 full rotation = 360 degrees = 360◦

I half rotation =

I quarter rotation =

I 1/8 rotation =

I 1◦ =

Right angle and Straight angle

Acute angle and obtuse angle

Coterminal angles

Angles that share the same initial and terminal sides are calledcoterminal angles.

List a few coterminal angles of the following angles1. −30◦

2. 370◦

Special Triangles

Examples

Radian measure of an angleHere we have a central circle of radius r .

Definition (central angle)

A central angle is an angle whose vertex is at the center of thecircle.

Definition (Radian measure of an angle)

If θ is a central angle in a circle of radius r , and is subtended by anarc length of s, then

θ =s

r

Examples

Radian measure of an angle (Continued)

So it’s a new way of defining angle measures. In words,

DefinitionRadian is the ratio between an arc length and its radius. That is,

Radian =arc length

radius(or, θ =

s

r)

Then,

I 1 radian is when arc length = radius

I 2 radian is when arc length = 2 times the radius

I etc...

Examples

Let the angle measure starts at P.

Example 1. r = 8cm, s = 4cm, what is θ in radians?Example 2. r = 8cm, s = 8cm, what is θ in radians?Example 3. r = 8cm, s = 16cm, what is θ in radians?

Special case: when we have a unit circle (i.e. r = 1 )

Radian =arc length

radius= arc length.

Thus, for the unit circle with r = 1,

I 1 radian is when arc length = 1

I 2 radian is when arc length = 2

I etc...

Remember π ∼ 3.14...?

What is π?

Definitionπ is the ratio of the circumference (c) to its diameter (2r).⇒ π = c

2r

Thus, for general r , c =

If r = 1, then what is c?

What does this mean?

2π = 360◦

What is 1 radian in terms of degrees?

What is 1 degree in terms of radians?

Special angles

Convert the following angles in degrees to radians.

I 0◦ =

I 30◦ =

I 45◦ =

I 60◦ =

I 90◦ =

I 180◦ =

I 270◦ =

I 360◦ =

Special angles (Continued)

Convert the following angles in radians to degrees.

I 0 =

I π6 =

I π4 =

I π3 =

I π2 =

More problemsConvert from the degree measures to radians (and vice versa).

θ = 210◦

θ = 2π3

θ = −3.7

Arc Length (s)Idea: Given a circle, an arc length can be figured out as long as Iknow the radius of the circle and the angle it subtends to.

Recall that θ in radians is given by the following:

θ =s

rwhere s = arc length and r = radius. Then,

s = rθ

Example 1. r = 10cm. Find s subtended by an angle of 3.5 rad.

Angular and Linear Velocity

Definition (Angular velocity)

The angular velocity (ω) is the amount of rotation per unit time.That is,

ω =θ

t

Ex 1) The round-a-bout makes 4/3 revolutions per 1 second.What is the angular velocity ω?Ex 2) The round-a-bout makes 1 revolution in 2 seconds. ω?

Linear Velocity

Definition (Linear velocity)

The linear velocity (V ) is the distance traveled by a point on thecircumference per unit time. That is,

V =s

t=

rθ

t= rω

Ex) Point P is rotating around the circumference of a circle withr = 2ft at a constant rate. If it takes 4 seconds for P to rotatethrough an angle of 360◦, what is the linear velocity of P?

Example

The wheels on a racing bicycle have a radius of 13 in. How fast isthe cyclist traveling in miles per hour (mph) if the wheels areturning at 300 rotations per minute (rpm)?(Note: 1 mile = 5280 ft = 5280 x 12 inches, and 1 min = 1/60 hr)

The area (A) of a circular sectorRecall that

I The area of a circle of radius r is given by πr2.

I one full revolution = 2π radians

Then, note that

Atotal area of circle

=angle subtended by arc s

angle subtended by circumference

=θ

2π

⇒ A = =1

2r2θ

ExampleIn the unit circle, what is the area of the quarter of the circle? halfthe circle? full circle?(Note multiple ways)

Example - Area of a slice of pizzaWhat is the area of one slice of the following pizza? The wholepizza, whose diameter is 14 inches, was cut equally into 6 pieces.

Review of Ch 6.1

DefinitionRadian is the ratio between an arc length and its radius. That is,

Radian =arc length

radius(or, θ =

s

r)

I From the above definition, we also have s = rθ.

Since2π = 360◦ ( or π = 180◦)

I To convert from degree measures to radian, multiply π180 .

I To convert from radian measures to degree, multiply 180π .

Review of Ch 6.1 (Continued)

Given a right triangle ∆ABC with θ as follows,

,the ratios of the three sides are;

I If θ = 30◦ (or π6 ) ⇒ b : a : c =

√3 : 1 : 2

I If θ = 45◦ (or π4 ) ⇒ b : a : c = 1 : 1 :

√2

I If θ = 60◦ (or π3 ) ⇒ b : a : c = 1 :

√3 : 2

Review of Ch 6.1 (Continued)

Definition (Angular velocity vs. Linear velocity)

The angular velocity (ω) is the amount of rotation per unit time:

ω =θ

t

The linear velocity (V ) is the distance traveled by a point on thecircumference per unit time:

V =s

t=

rθ

t= rω

Also, area of a sector of a circle: A = 12 r

2θ

HW for Ch. 6.1

Note: Show all work.1, 8, 12, 45, 46, 58, 64, 81, 83, 87