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