5.1 Angles and Their Measure

A unit circle is a circle with a radius of 1

It is centered at the origin on a coordinate plane

Angles in multiples of 30 and 45 degrees are included on the circle.

45°30°

60°90°120°135°

150°

180° 0°360°

210°225°

240°270°

300°315°

330°

5.1 Angles and Their Measure

Radian Measurements in a Circle

• What’s the circumference of a circle with a radius of 1?

5.1 Angles and Their Measure

radians

5.1 The Unit Circle

45°30°

60°90°120°135°

150°

180° 0°360°

210°225°

240°270°

300°315°

330°

Angles in multiples of and radians are included on the circle.

6

4

6

4

3

2

0

2

5.1 The Unit Circle

Finally, coordinate points on the circle are filled in.

Such as thepoints on each axis

)0,1(

)1,0(

)0,1(

)1,0(

5.1 The Unit Circle

)0,1(

)1,0(

)0,1(

)1,0(

?)(?,

?)(?,

?)(?,

How can we find the coordinates at 30 degrees? 45 degrees?

5.1 The Unit Circle

Two special triangles are used to coordinatize the unit circle

1)We use a 45-45-90 triangle with a hypoteneuse of 1

45°

45°

12

2

2

2

5.1 The Unit Circle

Two special triangles are used to coordinatize the unit circle

2) We also use a 30-60-90 triangle with a

hypoteneuse of 1

30°

60°

12

3

2

1

)0,1(

)1,0(

2

1,

2

3

5.1 The Unit Circle

1

2

3

2

1

30°

60°

)0,1(

)1,0(

2

2,

2

2

5.1 The Unit Circle

1

2

2

2

2

45°

45°

)0,1(

)1,0(

2

3,

2

1

5.1 The Unit Circle

1

2

1

2

3

60°

30°

)0,1(

)1,0(

)0,1(

5.1 The Unit Circle

2

1,

2

3

2

3

2

11

How do you fill in the rest of the coordinates? Ideas?

30°

60°

)0,1(

)1,0(

)0,1(

5.1 The Unit Circle

2

1,

2

3

2

3

2

11

Use reflections of the triangles and/or symmetry rules!

30°

60°

,

2

3

1

30°

60°

same y coordinatesopposite x coordinates

2

1

)0,1(

)1,0(

)0,1(

5.1 The Unit CircleHow do you fill in the rest of the coordinates? Use reflections of the triangles

,

2

2,

2

2

1

2

2

2

2

45°

45°

same y coordinatesopposite x coordinates

45°

45°

1

2

2

2

2

)0,1(

)1,0(

)0,1(

5.1 The Unit CircleHow do you fill in the rest of the coordinates?

2

3,

2

1

1

2

1

2

3

60°

30°

, same y coordinatesopposite x coordinates

5.1 The Unit Circle

45°30°

60°90°120°135°

150°

180° 0°360°

210°225°

240°270°

300°315°

330°

What your plate should look like when it’s complete…

6

4

3

2

3

2

3

4

02

3

5

4

3

4

54

7

6

5

6

7

6

11

2

3

2

1,

2

3

)0,1(

)1,0(

)0,1(

)1,0(

2

3,

2

1

2

2,

2

2

2

3,

2

1

2

3,

2

1

2

3,

2

1

2

2,

2

2

2

2,

2

2

2

2,

2

2

2

1,

2

3

2

1,

2

3

2

1,

2

3

5.1 The Unit Circle