Introduction to Radians and The Unit Circle

By: S.K.

Today you will learn...

How to find a fraction of a circle's circumference...

What π really is...

What a radian is... (hint: it's a unit of measure!)

What the unit circle is and why it is important...

Let's start with something you may already know...

What is the circumference (C) of a circle with radius (r) equal to 2 cm. ?

Take a couple minutes to work out the answer. You may work together or independently.

r = 2 cm.

C

Let's start with something you may already know...

What is the circumference (C) of a circle with radius (r) equal to 2 cm. ?

C = circumference, r = radius, d = diameter

Recall: C = πd = 2πr

Given: r = 2cm.

C = 2πr = 2π(2cm.) = 4π(cm.)

ANSWER: C = 4π(cm.)

What if we wanted to find a fraction of the circumference?Could we use the full circumference value to find it?What is 1/6 of the circumference (C) of a circle with radius (r) equal to

5 cm. ?

Take a couple minutes to work out the answer. You may work together or independently.

r = 5 cm.

C

What if we want to find a fraction of the circumference?Could we use the full circumference value to find it?What is 1/6th of the circumference (C) of a circle with radius (r) equal

to 5 cm. ?

C = circumference, r = radius, d = diameter

Recall: C = πd = 2πr

Given: r = 5cm. C = 2πr = 2π(5cm.) = 10π(cm.)

Since we have now have the full circumference, we can solve for 1/6th of the circumference by multiplying...

(1/6)(10πcm.) = (5π/3)(cm.)

ANSWER: C = (5π/3)(cm.)Or we can simply integrate the fraction into the original formula...

(1/6)C = (1/6)2πr = (π/3)(5cm.) = (5π/3)(cm.)

Amended Formula: (fraction)C = (fraction)πd = (fraction)2πr

Let's talk about π(pi)... You have most likely been using π in your math classes for a couple of years

now, most notably in formulas for circumference (C=2πr) and area (A=πr ) of a circle.

We know that π is approximately equal to 3.14159...

What do you think π really is? What is π a measure of? What is the significance of the value of π? Ponder these questions for a couple minutes. You may confer with your fellow classmates. Fill in your answers on your guided notes.

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Let's talk about π(pi) and The Unit Circle... π arises when we study one of the most important tools of trig., The Unit Circle!!! The Unit Circle is a circle of radius equal to 1 (r=1), centered at the origin (0,0). If we were to split up the circumference/arc-length of half of the unit circle circle (180 ) into π equal sized chunks, we call each of these chunks a radian. Therefore, we can say that a 180 arc-length of the unit circle is equal to π radians, and

the full circle (360 ) is equal to 2π radians.

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Let's talk about π(pi) and The Unit Circle... We can further subdivide these using fractions to come up with a

radian equivalent to any degree measure, the same way we use fractions to find a piece of the circumference!!!

This gives us a new way to look at angles!!! Example 1: Convert 40 degrees to radians...

We know: π = 180 degrees (half the circle) (40/180) = (2/9)

Since (2/9) is a proportion of half the circle we multiply this proportion by the number of radians in half the circle (π) to get the radian measure of 40 ...

(2/9) * π radians = (2π/9) radians

Answer: 40 degrees = (2π/9) radians

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Examples: Convert degree to radian... Now you try!!!

Example 2: Convert 50 degrees to radians...

Example 3: Convert 60 degrees to radians...

Take a few minutes to figure these out. Refer to your notes and Example 1. You may work together with other students or independently. Fill in your solutions on your guided notes.

Examples: Convert degree to radian... Example 2: Convert 50 to radians...

Example 3: Convert 60 to radians...We know: π = 180 degrees (half the circle) (60/180) = (1/3)

Since (1/3) is a proportion of half the circle we multiply this proportion by the number of radians in half the circle (π) to get the radian measure of 60 degrees...

(1/3) * π radians = (π/3) radians

Answer: 60 degrees = (π/3) radians

We know: π = 180 degrees (half the circle) (50/180) = (5/18)

Since (5/18) is a proportion of half the circle we multiply this proportion by the number of radians in half the circle (π) to get the radian measure of 50 degrees ...

(5/18) * π radians = (5π/18) radians

Answer: 50 degrees = (5π/18) radians

Questions to consider...

Why do you think we use radians?

What benefits might radians have over degrees?

Why do you think we study trigonometry using the unit circle?

(Ponder these questions, and put your responses in your guided notes)

Video 1: Radians & Degrees (by: Khan Academy)

Radians & Degrees (by: Khan Academy) (Click Here)(https://www.youtube.com/watch?&v=9zspW8u6kQM)

Watch the video, and fill in your guided notes!!!

Video 2: The Unit Circle (by: Khan Academy)

This video will give you a preview of where the class is going next...

Introduction To The Unit Circle (by: Khan Academy) (Click Here)(https://www.youtube.com/watch?&v=1m9p9iubMLU)

Watch the video, and fill in your guided notes!!!

And now a song...

Here is a song to help you remember what you have learned today!

https://www.youtube.com/watch?v=1CiXAP8XaBg (if song doesn't play click this link)