Download r = π 180 ⋅d To convert from radians to degrees: d = 180 π ⋅r S

Scholarship Algebra II
Circular Trig Section 1: Angles and Radians
We often name angles with Greek letters – the most common are theta (θ) and alpha (α). You
can use other Greek letters if you get bored with these.
Circular Trigonometry: We will now talk about angles on a circle instead of angles in triangles.
This is called standard position.
The initial side is always the positive x-axis.
initial side
The terminal side is the ray where the angle
ends.
The sign of the angle will indicate direction.
Positive angles go counter-clockwise, and
negative angles go clockwise.
terminal side
The positive x-axis is 0 degrees.
Coterminal Angles are angles with the same terminal side.
The angle at the left could have several different
measures.
160°
-200°
520°
Radians
A radian is another way to measure angles. One radian is one radius unit around the circle.
An entire circle contains 2π radians (equivalent to 360°).
r 2π
π
=
=
d 360 180
π
⋅d
180
180
To convert from radians to degrees: d =
⋅r
π
To convert from degrees to radians: r =
Ex. 1: Convert to radians.
a) 60°
π
πR
60 ⋅
=
180 3
c) -300°
π
5π R
−300 ⋅
=−
180
3
b) 225°
π
5π R
225 ⋅
=
180
4
d) 144°
π
4π R
144 ⋅
=
180
5
Ex. 2: Convert to degrees.
5π R
a)
3
5π 180
⋅
= 300°
3 π
3π R
b)
2
3π 180
⋅
= 270°
2 π
7π R
c) −
12
−
7π 180
⋅
= −105°
12 π
Arc Length
How would we find the arc length of a circle with a radius of 4 ft and a central angle of 75°?
75
5
=
360 24
What’s the length around the whole circle? The circumference! C = 2π r = 8π
What fraction of the circle is 75°?
So the arc length is: s =
5
5π
( 8π ) = ≈ 5.24 ft
24
3
Arc Length Formulas:
Degrees: s =
θ
⋅ 2π r
360
θ
⋅ 2π r
2π
s = θr
Radians: s =
5π R
Ex. 4: r = 4 cm, θ =
6
Find arc length.
Ex. 3: r = 2.8 cm, central angle = 330°
Find arc length.
s=
⎛ 5π ⎞ 10π
s = rθ = 4 ⎜ ⎟ =
≈ 10.5 cm
⎝ 6 ⎠
3
330
⋅ 2π ( 2.8 ) ≈ 16.1 cm
360
 = 35π , α = 315° Find r.
Ex. 5: mAB
s=
θ
⋅ 2π r
360
35π =
315
⋅ 2π r
360
r = 35 ⋅
360
= 20
630