5.1 Angles and Degree Measure

Angle- formed by rotating a ray about its endpoint (vertex)

Initial Side Starting position

Terminal Side Ending position

Standard PositionInitial side on positive x-axis and the vertex is on the origin

Angle describes the amount and direction of rotation

120° –210°

Positive Angle- rotates counter-clockwise (CCW)

Negative Angle- rotates clockwise (CW)

An angle is formed by joining the endpoints of two half-lines called rays.

The side you measure from is called the initial side.

Initial Side

The side you measure to is called the terminal side.

Terminal S

ide

This is a counterclockwise rotation.

This is a clockwise rotation.

Angles measured counterclockwise are given a positive sign and angles measured clockwise are given a negative sign.

Positive Angle

Negative Angle

• Standard Position:  An angle is in standard position if its vertex is located at the origin and one ray is on the positive x-axis

• Angle Measurement• The two most common units of

measurement for angles are degrees and radians. There are two variations used in the “degree system” – here we describe only the one called degreedecimal.

• In the degree decimal system…• 1 complete revolution = 360 degrees.

Degree Measure

Over 2500 years ago, the Babylonians used a number system based on 60

The number system we use today is based on 10

However we still use the Babylonian idea to measure certain things such as time and angles. That is why there are 60 minutes in an hour and 60 seconds in a minute.

Degree Measurement and the Babylonians

• In a full circle there are 360 degrees

Each degree is split up into 60 parts, each part being 1/60 of a degree.  These parts are called minutes

Each minute is split up into 60 parts, each part being 1/60 of a minute. These parts are called seconds

Converting …say 121.135 degrees

• The whole units of degrees will remain the same (i.e. in 121.135° longitude, start with 121°).

• Multiply the decimal by 60 (i.e. .135 * 60 = 8.1).• The whole number becomes the minutes (8').• Take the remaining decimal and multiply by 60.

(i.e. .1 * 60 = 6).• The resulting number becomes the seconds (6").

Seconds can remain as a decimal.• Take your three sets of numbers and put them

together, using the symbols for degrees (°), minutes (‘), and seconds (") (i.e. 121°8'6" longitude)

Convert 112.420 to DMS

Convert the fractional part

'2.256042. Convert the fractional part of the minutes into seconds

''12602. '''00 122511242.112

Let’s look at the special angles called the quadrantal angles.

90

180

270

0

The quadrantal angles are those angles that lie on the axis of the Cartesian coordinate system: , , , and .0 90 180 270

• Ex: Give the measure of the angle represented by 3.3 rotations counterclockwise.

Coterminal Angles

• Angles with the same initial side and same terminal side, but have different rotations, are called coterminal angles.

• 50° and 410° are coterminal angles. Their measures differ by a multiple of 360.

Q: Can we ever rotate the initial side counterclockwise more than one revolution?

Answer – YES!

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Note: Complete Revolutions

Rotating the initial side counter-clockwise

1 rev., 2 revs., 3revs., . . .

generates the angles which measure

360, 720, 1080, . . .

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Picture

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ANGLES 360, 720, & 1080 ARE ALL COTERMINAL

ANGLES!

What if we start at 30 and now rotate our terminal side counter-clockwise 1 rev., 2 revs., or 3 revs.

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• Suppose we are given an angle measure of 775 degrees.

• What would the angle measure be of its terminal side?

If A is an angle in standard positon, its reference angle Ar is the acute angle formed by the x axis and the terminal side of angle

A

Reference angle rule

Based on the location of the terminal side…

• Ex: Find the measure of the reference angle for -135 degrees.