Download 4.2 The Unit Circle and Reference Angles

PREC 12
4.2 The Unit Circle and Reference Angles
Date:
Complete the following table by converting each to degrees. Note the patterns that exist!
Arc Length and Sector Angles
One of the advantages to measuring angles in radians is for calculating arc lengths. There is a close relation between the
arc length and the size of the angle.
The general relation can be summarized
with the following formula:
a = rθ
arc length
sector angle
=
circumference
angle of circle
arc length
sector angle
=
2π r
2π
arc length =
2π r × sector angle
2π
a = arc length
r = radius
θ = sector angle
Note: the sector angle must be
measured in radians. If it is provided in
degrees, you must convert it to units of
radian.
Example 1: An arc subtends an angle of 1.8 rad. If the radius of the circle is 6 cm, what is the arc length?
Example 2: An arc subtends the angle θ . If the radius is 18 cm and the arc length is 90 cm, what is the measure of angle
θ to the nearest tenth of a radian?
Example 3: Determine the length of the arc of a circle with radius 12m that subtends 240° at the centre. Express the
length to 2 decimal places.
Pythagorean Theorem and Special Triangles
Ratio of Sides:
3-4-5 Right Angle Triangle
Ratio of Sides:
5-12-13 Right Angle Triangle
Ratio of Sides:
Right Angle Isosceles
45o- 45o - 90o
Triangle
Ratio of Sides:
Half an Equilateral
30o - 60o - 90o
Triangle
Let’s look closely in the first quadrant when we incorporate the special triangles in a UNIT CIRCLE:
30° or
π
6
reference angle in Quad 1
What is the coordinate at 0° or 0π ?
45° or
π
4
reference angle in Quad 1
60° or
π
3
reference angle in Quad 1
And the coordinate at 90° or
π
2
?
Example 4: Determine the point on the terminal arm that intersects the unit circle given P (θ ) :
a. P ( 76π ) in standard position
b. P ( − π4 ) in standard position
Example 5: Determine the reference angle and the angle in standard position, in radians, given the following
coordinates:
a.
(−
2
2
,
2
2
)
b.
(
1
2
,−
3
2
)
Example 6: Determine the reference angle if the angle in standard position is 5.26 radians.