Download Trigonometry

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
r
r
Definition: One
Radian is the
measure of a central
angle of a circle that
is subtended by an
arc whose length = r.
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.
This sector = 3r. Therefore, the
central angle = 3 radians.
r
r
r
r
s =  ·r
s
Therefore q =
r
arc lenth
central angle =
radius
What is the radian measure of a
whole circle?
For a whole circle, s = circumference = 2p r
So, qwhole
circle
s 2p r = 2p
= =
r
r
What does this mean? We now have a
conversion factor (dimensional analysis)
2p = 360°
or
p = 180°
Why do this?
Examples:
1. Convert 55 to radians.
5p
2. Convert
to degrees.
4
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
2
r
k=
2
2
p r = k ×2p
2
rq
A sec tor =
2
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
A sec tor
x
r (x) p
2
=
pr
=
×
360
2 180
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.