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Trigonometry, Pt 1: Angles and Their
Measure Mr. Velazquez
Honors Precalculus
Defining Angles • An angle is formed by two rays or segments that intersect at a common
endpoint.
• One side of the angle is called the initial side, and the other is called the terminal side.
• For convenience, it’s useful to think of an angle as a stationary initial side with a terminal side that rotates around it, with counterclockwise rotation indicating a positive angle and clockwise rotation indicating a negative angle.
• An angle is in standard position if: • Its vertex is at the origin of the coordinate system • Its initial side lies along the positive x-axis
Measuring Angles in Degrees
Measuring Angles in Radians
Measuring Angles in Radians
EXAMPLE What is the radian measure 𝜃 for an arc of length 15 inches and a radius of 6 inches?
Conversion Between Degrees and Radians
The arc length of a full circle (360°) is essentially the entire circumference of the circle. This angle is therefore equal to 𝟐𝝅 radians. A half circle has an angle measure equal to 𝝅 radians.
Conversion Between Degrees and Radians
EXAMPLE
Convert the following angles in degrees into radians.
a) 135°
b) –45°
c) 60°
d) –120°
Conversion Between Degrees and Radians
EXAMPLE
Convert the following angles in radians into degrees.
a)𝜋
2
b) −𝜋
c)5𝜋
3
d) −𝜋
6
Angles in Standard Position
Often, we can get a sense of where an angle is located based on certain reference angles. A few of these reference angles are given below:
Angles in Standard Position EXAMPLE Draw and label each angle in standard position:
a)α =3π
2
b)β = 2π
c)θ =7π
4
Angles in Standard Position
Below are select positive and negative angles, given in radians and degrees.
A table showing the same standard angle measures and their conversions to radian and degrees. BTW: 1 revolution (equal to 360° or 2𝜋 radians) is often used as a unit of angle measurement in science and technology.
Coterminal Angles
Notice that the angle measurements 90° and −270° both refer to the same exact angle. This means 90° and −270° are coterminal angles.
Any angle 𝜃 is coterminal with angles of:
Where 𝑘 is any integer.
𝜃 + 𝑘 ⋅ 360°
Coterminal Angles
EXAMPLE
Find a positive angle less than 360° that is coterminal with each of the following:
a) 390°
b) 405°
c) –135°
Coterminal Angles
EXAMPLE
Find a positive angle less than 2𝜋 radians that is coterminal with each of the following:
a)5π
2
b)11π
4
c) −π
6
Length of a Circular Arc
EXAMPLE A circle has a radius of 7 inches. Find the length of the arc intercepted by a central angle of 120°.
Linear and Angular Speed
Linear and Angular Speed
EXAMPLE
A bicycle tire with a radius of 80 cm rotates with an angular speed of 3𝜋 radians per second. A piece of gum is stuck to the edge of the tire. What is the linear speed of the piece of gum, in cm/s?
Exit Ticket: Angles A windmill is used to generate electricity. Its blades are 12 feet in length, and rotate at an angular speed of 8 revolutions per minute. Find: a) The linear speed at the tips of the blades,
in ft/s. b) The central angle (in radians and
degrees) each blade will spin through in 3 seconds.
Homework: Pg 472-473, #4-80
(multiples of 4)
Remember: Linear Speed 𝑣 =
𝑠
𝑡
Angular Speed 𝜔 =𝜃
𝑡
Arc Length 𝑠 = 𝑟𝜃