3-Unit-Circle-Angles-and-their-Measuretrigo.pdf

Angles in the unit circle, special angles, their measurements

1. 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

2. 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

3. Radian Measurements in a Circle
•What’s the circumference of a circle with a
radius of 1?
5.1 Angles and Their Measure
radians

4. 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
p
4
p
6
p4
p
3
p2
p
p p0
p2

5. 5.1 The Unit Circle
Finally, coordinate points on the circle are filled in.
Such as the
points on
each axis
)0,1(
)1,0(
)0,1(-
)1,0(-

6. 5.1 The Unit Circle
)0,1(
)1,0(
)0,1(-
)1,0(-
?)(?,
?)(?,
?)(?,
How can we find the coordinates at 30 degrees? 45 degrees?

7. 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°
1
2
2
2
2

8. 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°
1
2
3
2
1

9. )0,1(
)1,0(
÷
÷
ø
ö
ç
ç
è
æ
2
1
,
2
3
5.1 The Unit Circle
1
2
3
2
1
30°
60°

10. )0,1(
)1,0(
÷
÷
ø
ö
ç
ç
è
æ
2
2
,
2
2
5.1 The Unit Circle
1
2
2
2
2
45°
45°

11. )0,1(
)1,0(
÷
÷
ø
ö
ç
ç
è
æ
2
3
,
2
1
5.1 The Unit Circle
1
2
1
2
3
60°
30°

12. )0,1(
)1,0(
)0,1(-
5.1 The Unit Circle
÷
÷
ø
ö
ç
ç
è
æ
2
1
,
2
3
2
3
2
11
How do you fill in the rest of the coordinates? Ideas?
30°
60°

13. )0,1(
)1,0(
)0,1(-
5.1 The Unit Circle
÷
÷
ø
ö
ç
ç
è
æ
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 coordinates
opposite x coordinates
2
1

14. )0,1(
)1,0(
)0,1(-
5.1 The Unit Circle
How do you fill in the rest of the coordinates?
Use reflections of the triangles
( ) , ÷
÷
ø
ö
ç
ç
è
æ
2
2
,
2
2
1
2
2
2
2
45°
45°
same y coordinates
opposite x coordinates
45°
45°
1
2
2
2
2

15. )0,1(
)1,0(
)0,1(-
5.1 The Unit Circle
How do you fill in the rest of the coordinates?
÷
÷
ø
ö
ç
ç
è
æ
2
3
,
2
1
1
2
1
2
3
60°
30°
( ) ,
same y coordinates
opposite x coordinates

16. 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
p
4
p3
p
2
p
3
2p
3
4p
p p0
p2
3
5p
4
3p
4
5p
4
7p
6
5p
6
7p
6
11p
2
3p
÷
÷
ø
ö
ç
ç
è
æ
2
1
,
2
3
)0,1(
)1,0(
)0,1(-
)1,0(-
÷
÷
ø
ö
ç
ç
è
æ
2
3
,
2
1
÷
÷
ø
ö
ç
ç
è
æ
2
2
,
2
2
÷
÷
ø
ö
ç
ç
è
æ
-
2
3
,
2
1
÷
÷
ø
ö
ç
ç
è
æ
--
2
3
,
2
1
÷
÷
ø
ö
ç
ç
è
æ
-
2
3
,
2
1
÷
÷
ø
ö
ç
ç
è
æ
-
2
2
,
2
2
÷
÷
ø
ö
ç
ç
è
æ
--
2
2
,
2
2
÷
÷
ø
ö
ç
ç
è
æ
-
2
2
,
2
2
÷
÷
ø
ö
ç
ç
è
æ
-
2
1
,
2
3
÷
÷
ø
ö
ç
ç
è
æ
--
2
1
,
2
3
÷
÷
ø
ö
ç
ç
è
æ
-
2
1
,
2
3

17. 5.1 The Unit Circle