Download Math 1316 – Trigonometry Section 1.1 Radian and Degree Measure

Math 1316 – Trigonometry
Section 1.1 Radian and Degree Measure
An angle is a rotation of ray R1 onto R2 .
If the rotation is counterclockwise, the angle is
is
.
. If the rotation is clockwise, the angle
The measure of an angle is the amount of rotation about the vertex from the initial side to the terminal side.
Can measure angles in degrees or radians.
1◦ is equivalent to rotating one side
1
of a complete revolution.
360
If a circle of radius 1 is drawn with the vertex of an angle at its center, then its measure in radians is the length
of the arc that subtends the angle.
In degrees, 1 revolution is
Relationship between degrees and radians:
. In radians, one revolution is
.
1. To convert from degrees to radians, multiply by
2. To convert from radians to degrees, multiply by
NOTE: 1 rad ≈
Ex: (a) Express 60◦ in radians.
Ex: Convert 15◦ 150 4200 to radians.
1◦ ≈
(b) Express
π
rad in degrees.
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1 ◦
(1 )
60
1
100 =
(1◦ )
3600
10 =
Math 1316
Section 1.1 Continued
An angle is in
at the origin and the initial side on the positive x−axis.
Two angles in standard position are
if it is drawn in the xy−plane with the vertex
if their sides coincide.
Ex: Find angles coterminal with θ in standard position.
(a) θ = 130◦
(b) θ =
19π
3
π
are complementary. Two positive angles whose sum is π are supplementary.
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Ex: Find the complement and supplement of the following angles:
Two positive angles whose sum is
(a)
11π
12
(b)
(c) 18◦
5π
8
(d) 130◦
Arc Length In a circle of radius r, the length s of an arc that subtends a central angle of θ radians is
NOTE:
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Math 1316
Section 1.1 Continued
We can now define radian measure using circles of any radius (not just r = 1).
Thus, we can define a radian to be the measure of a central angle θ that intercepts an arc equal in length to the
s
radius of the circle. Algebraically, θ = where θ is measured in radians.
r
NOTE: The arc length formulas are only true when θ is in
Ex: A circular arc of length 3 ft subtends a central angle of 35◦ . Find the radius of the circle.
Area of a Sector of a Circle: In a circle of radius r, the area A of a sector with central angle of θ radians is
Ex: A sector of a circle has a central angle of 60◦ . Find the area of the sector if the radius is 5 miles.
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Math 1316
Section 1.1 Continued
Circular Motion: Suppose a point moves along a circle of radius r and the ray from the center of the circle to
the point traverses θ radians in time. Let s = rθ be the distance the point travels in time t. The speed of the
object is given by
Angular speed:
Linear speed:
If a point moves along a circle of radius r with angular speed ω, then its linear speed is given by
Ex: A ceiling fan with 16 in blades rotates at 45 rpm. Find the angular speed of the fan in rad/min. Find the
linear speed of the tips of the blades in inches/min.
Ex: A truck with 48 inch diameter wheels is traveling at 50 mi/h. Find the angular speed of the wheels in
rad/min. How many revolutions per minute do the wheels make?
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