Introduction to Radian Measure

MHF4UIFriday November 2nd, 2012

History of Degree Measurement

Before numbers and language ancient civilizations used the stars and constellations, or astronomy, to mark the seasons and

predict the future.

These ancient civilizations noticed that recognizable constellations would circle the skies above them over the course

of the four seasons.

They established that the transition through four seasons, or a year, had 360 days. Which is why we measure a circle to have 360

degrees.

History of Radian Measurement

The history of Radian measurement begins in the 1700s.

A mathematician named Roger Cotes discovered the relationship between sine and cosine functions.

During the same time period Leonard Euler explicitly said to measure angles by the length of the arc cut off in the unit circle

The word "radian" was coined by Thomas Muir and/or James Thompson about 1870, even though mathematicians had been

measuring angles that way for a long time.

What is a Radian?

Much like a degree, a Radian is a measurement of an angle.

The radian measure of an angle ,ϴ, is defined as the length, a, of the arc that “subtends” the angle divided by the radius of the arc ,r

𝜃=𝑎𝑟

How many Radians are in one Revolution?

In a circle we measure the circumference as:

The circumference of a circle is equivalent to one complete revolution around the circle.

Therefore one complete revolution measures Radians.

Radian Relationship to DegreesWe just established that one revolution around a circle is .

In degree measurement this is equivalent to 360

Radian Relationship to Degrees (Continued)

Let’s know take a look at how many Radians are in a Degree:

Example Conversion ProblemsExample 1:

Determine an exact and an approximate radian measure for an angle of 60

Example Conversion ProblemsExample 2:

Determine a degree measure for an angle of

Example Conversion ProblemsExample 3:

Determine a degree measure for an angle of 4.56

What would the new title be in Radians?

Finding Arc Length (a)

When asked to find the arc length, or some distance travelled around an arc you must use the formula:

Where the radian measure of an angle ,ϴ, is defined as the length, a, of the arc that “subtends” the angle divided

by the radius of the arc ,r

Solving for a

Example Arc Length ProblemYou made a trip to Canada’s Wonderland during the summer and you

choose to ride on the Carousel because it is your favorite ride. You choose to ride on a horse that is 10.4 metres from the centre of the

carousel.If the carrousel turns through an angle of , determine the length of

the arc that you just travelled in metres.

Finding Angular VelocityAngular velocity is the rate at which the central angle

changes over time.

RPM or Revolutions per Minute is a great example of angular velocity.

Situations where you encounter angular velocity:RPM of a car to change gears

The speed of a satellite orbiting the Earth

Dining in the restaurant of the CN tower

Example: Finding Angular VelocityWhen driving a car you shift gears at around 3,500 rpm.

Determine the exact angular velocity of the crankshaft at the time you are shifting gears in:

a) Degrees per secondb) Radians per second

Example: Finding Angular VelocityWhen driving a car you shift gears at around 3,500 rpm.

Determine the exact angular velocity of the crankshaft at the time you are shifting gears in:

a) Degrees per secondb) Radians per second

Homework Questions:

• Textbook Chapter 4.1 (Page 208)• Part A: 1,5,6,7,8, 9• Part B: 10, 15, 16, 17