Why do we use angles?Here is the theory…

Ancient Babylonians measured the path of the stars from night to night and noticed that they traveled in a circle over one year.

What does the night sky and a year have to do with angles?

Why do degrees measure 360 and not 365 ¼?

360 also has the factors 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 60….

Ancient Babylonians used a number system based on 60.

7.1 Measurement of Angles

I will be able to measure central angles using radians and convert

between degrees and radians

Objective

Angles of the Coordinate System

An angle (θ) is in standard position if• its vertex is at the origin of a coordinate system • its initial side lies along the positive x-axis.

x

y

Terminal Side

Initial SideVertex

is positive

Positive angles rotate counterclockwise.

x

y

Terminal Side

Initial SideVertex

is negative

Negative angles rotate clockwise.

Special AnglesReference Angles are acute angles

formed by the terminal side of the

given angle and the x-axis.

Reference Angles aid in graphing.

Coterminal angles are two angles that share the same terminal ray.They should also share the samereference angle when graphing.

Graph the following angles in standard position. Then find one positive and one negative coterminal angle for each. a. 20º angle b. –120º angle.

Examples

Find the reference angle for each. a. 120º angle b. 225º angle.

Examples- find two angles, one positive and one negative, that are coterminal with each given angle.

Then, state the reference angle.

1. 10°

2. 100°

3. -5°

4. 400°

1. 370°,-350°Ref: 10°

2. 460°,-260°Ref: 80°

3. 355°,-365°Ref: 5°

4. 40°,-320°Ref: 40°

So what is wrong with degrees then?

We could divide a circle into anything…

If we lived on Mars then maybe we would have 680o instead of 360o. (This is the number of days in Mars year.)

We have a second system to measure angles that has more to do with a circle than the days

in a year.

We call these radians.

Cut the pipe cleaner the length of the radius of your circle. Lay the radius pipe cleaner along your circle and mark the length.Repeat until you get all the way around the circle.

A radian is an arc length of the circle that equals the length of the radius.

Lay your pipe cleaner along the remaining portion that didn’t make a whole radius. Bend the pipe cleaner to measure that length.

Find what percent the remaining portion is to a radius.

How many pipe cleaners did it take to cover the entire circumference of the circle?

Have we seen this number before?

The exact number is 6.283185307…

2π

So, in a full circle, we have360⁰ = 2π radians

How many radians are in half a circle (180⁰)?

How many radians are in 1⁰?

How many degrees are in 1 radian?

Convert each to either radians or degrees

• 1. Convert 196° to radians (nearest hundredth)

• 2. Convert 1.35 radians to degrees (nearest tenth)

• 3. Convert 240° to radians- leave answer in terms of π

• 4. Convert to degrees

Homework

• Page 261 #3, 7, 18 GRAPH TOO!

• Fill out Unit Circle organizer with radians

Exit Slip

1. Convert the following into degrees or radians.

a. b. 226˚

2. Find a positive and negative coterminal angle for the following:

410˚