Lesson 7-1

Angles, Arcs, and Sectors

Objective:Objective:

Objective:Objective:

• To find the measure of an angle in either degrees or radians and to find coterminal angles.

An angle is made up of two An angle is made up of two rays:rays:

An angle is made up of two An angle is made up of two rays:rays:

Initial Ray

An angle is made up of two An angle is made up of two rays:rays:

Initial Ray

Terminal R

ay

An angle is made up of two An angle is made up of two rays:rays:

Initial Ray

Terminal R

ay

The angle measure (or opening between the two rays) we define as θ.

θ

An angle is made up of two An angle is made up of two rays:rays:

Initial Ray

Terminal R

ay

Our common unit for measuring smaller angles is the degree.

θ

• We all know there is 180° in a triangle and we also should all know by now there is 360° in a circle.

• A circle could just represent one complete revolution about a point.

• A circle could just represent one complete revolution about a point.

• So therefore in a revolution we say there is 360°.

• For angles that are very large, we may measure them in revolutions instead of degrees.

• For angles that are very large, we may measure them in revolutions instead of degrees.

• So, 5 revolutions= 5(360) = 1800°.

• Now, degrees can be broken down into smaller units.

• Now, degrees can be broken down into smaller units.

• For instance every degree has 60 minutes in its unit or 1° = 60 minutes.

• Just like in a clock each minute can be broken down into smaller units called seconds.

• How many seconds do you think are in 1 minute (denoted 1’)?

• How many seconds do you think are in 1 minute (denoted 1’)?

• 60 seconds – or 60” = 1’

• A measurement that looks like this 25° 20’ 6” is read as 25 degrees, 20 minutes, 6 seconds.

• Convert this measurement 25° 20’ 6”.

• Convert this measurement 25° 20’ 6”.

• Convert this measurement 25° 20’ 6”.

• Convert this measurement 12.3°.

• Convert this measurement 12.3°.

Another unit of measure for angle is called the radian::

Another unit of measure for angle is called the radian::

Another unit of measure for angle is called the radian::

Central Angle

Another unit of measure for angle is called the radian::

The vertex of the angle is located at the center of a circle.

Central Angle

Another unit of measure for angle is called the radian::

The rays of the angle are both radii of the circle.

Central Angle

Another unit of measure for angle is called the radian::

Traditionally call a central angle’s measure θ.

Central Angle

Another unit of measure for angle is called the radian::

The piece of the circle that is located between the two rays of the circle is called the arc length

of the central angle.

Another unit of measure for angle is called the radian::

In general, the radian measure of a central angle is the number of radius units in the length of the intercepted arc of the circle between the

two rays.

Another unit of measure for angle is called the radian::

Arc length is always denoted as s.

Another unit of measure for angle is called the radian::

Radius is always denoted as r.

Another unit of measure for angle is called the radian::

The measure of the central angle is always θ.

Another unit of measure for angle is called the radian::

This formula is always true s = rθ so,

One important conversion from radians to degrees is

One important conversion from radians to degrees is

From this we can

come up with these two

equivalences:

From this we can

come up with these two

equivalences:

From this we can

come up with these two

equivalences:

Convert 196° to radians (to the nearest hundredth).

Convert 1.35 radians to decimal degrees (to the nearest tenth) and to degrees and minutes (to the nearest ten minutes).

Angle measures are often given in terms of π.

Angle measures are often given in terms of π.

Some common angles and their conversion in terms of

radians:

Angle measures are often given in terms of π.

Some common angles and their conversion in terms of

radians:

Angle measures are often given in terms of π.

Some common angles and their conversion in terms of

radians:

Angle measures are often given in terms of π.

Some common angles and their conversion in terms of

radians:

• If an angle is in standard position, then the vertex is located at the origin.

• The initial ray is located on the positive x-axis.

• The terminal ray is where the angle of rotation stops and it can be located anywhere.

• If the terminal ray lies on an axis then the angle is called a quadrantal angle.

• If the terminal ray lies on an axis then the angle is called a quadrantal angle.

• This will always be a multiple of 90 or .

• Co-terminal angles are any pair of angle that share the same terminal ray.

• Co-terminal angles are any pair of angle that share the same terminal ray.

• For any given angle there are infinitely many co-terminal angles.

Find two angles, one positive and one negative

that are co-terminal with the angle π/4.

Find two angles, one positive and one negative

that are co-terminal with the angle π/4.

Show a sketch!

Assignment:

Pgs. 260 - 261 C.E. 1-6 all

W.E.1-31 odd, 32