6-1 Angle Measures

The Beginning of Trigonometry

New Way of Thinking

We were used to drawing figures in an x/y plane and looking at their relationships. Now we are going to base everything on a circle drawn on the x/y plane. We’ll be looking at triangles created and the relationships of the sides of the triangles. This, my friends, is trigonometry!!

The Radian

The Radian is a new way to measure an angle. We are used to using degrees to indicate the size of an angle. Now we will use this new measuring tool.

Its like being in Canada - the kilometer vs. the mile. The radian vs. the degree - either way, it indicates the specific size of an angle.

The Radian

Definition: One Radian is the measure of a central angle of a circle that is subtended by an arc whose length = r.

r

r

One radian

Some Variables commonly used

Arc Length = s

Central Angle =

Radius = r

Now we’re going to see how these relate

I’m going to draw a picture with a central angle = 3 radians, then find the relationship between arc length, central angle, and radius.

r

r

r

r

This sector = 3r. Therefore, the central angle = 3 radians.

s = ·r

Therefore

sr

q=

arc lenthcentral angle

radius=

What is the radian measure of a whole circle?

For a whole circle, s = circumference =

So,

What does this mean? We now have a conversion factor (dimensional analysis)

2 rp

2 rrp

=wholecircle

sr

q = 2= p

2 360p= ° or 180p= °

Why do this?

Examples:

1. Convert 55 to radians.

2. Convert to degrees.54p

Now what?

Do you remember areas of sectors from Geometry?

Essentially, a sector is a piece of pizza. The area is dependant on the central

angle right? For any particular circle, the radius is

constant. Only the central angle changes. So…

A k= ×q 2r k 2p = ×p

2rk

2=

2

sector

rA

2q

=

Note: central angle should be in radians!!

What would the formula be if the angle was in degrees?

You’d need to include the conversion factor 2

sector

r (x)A

2 180p

= × 2xr

360= p

Remember?

Examples

3. Find the area of the sector of a circle with radius 4 if the central angle = 120.

4. Find the area of the sector of a circle with radius 2 if the central angle =

5. Find the area of the sector of a circle with radius 2 if the subtended arc = 6.