1. Angles and Their Measures
Chapter 4, Sections 1 & 3

2. Angles
An angle is formed by two rays that have a common endpoint called the vertex.
One ray is called the initial side and the other the terminal side.
The arrow near the vertex shows the direction and the amount of rotation from the initial side to the terminal side. A è B C
Terminal Side
Initial Side
Vertex

3. Standard Position An angle is in standard position if
its vertex is at the origin of a rectangular coordinate system and
its initial side lies along the positive x-axis.

4. Standard Position – Positive Anglesx y
Terminal Side
Initial Side
Vertex 
 is positive
Positive angles rotate counterclockwise.

5. Standard Position – Negative Anglesx y
Terminal Side
Initial Side
Vertex 
 is negative
Negative angles rotate clockwise.

6. Measuring Angles Radians
We measure angles in 2 different units: degrees and radians
Degrees
Divided into 60 equal parts called minutes (‘)
Divided into 60 equal parts called seconds (“)
Radians
Uses π

7. Angle Conversions – Degrees to DMS
Keep the whole number – this is the degree part (ᵒ)
Multiply the decimal part by 60 – this is the minutes part (‘)
Multiply the remaining decimal part by 60 – this is the seconds part (“)

8. Angle Conversions – DMS to Degrees
Keep the degree part – this is the whole number.
Divide the minutes part by 60, divide the seconds part by 3600, then add those two numbers to each other – this is the decimal.

9. Angle Conversions – Degrees to Radians
Divide the degrees by 180
Multiply by π
**leave in fraction form**
**if the angle is in DMS form, convert back to degrees first**

10. Angle Conversions – Radians to Degrees
Divide by π
Multiply by 180

11. Special Angles – Coterminal Angles
two angles that share a terminal side
To find coterminal angles – add or subtract any number of full circles to the angle
The degree measure of an angle has been increased/decreased by a multiple of 360º
The radian measure of an angle has been increased/decreased by a multiple of 2π

12. Special Angles – Reference Angles
the acute angle (A) formed by the terminal side of the given angle and the x-axis
In Quadrant I, the reference angle is A
In Quadrant II, the reference angle is 180-A
In Quadrant III, the reference angle is A-180
In Quadrant IV, the reference angle is 360-A

13. Special Angles – Reference Angles

14. Special Angles – Quadrantal Angles
the terminal side of the angle coincides with one of the axes
90º
180º
270º
360º

15. In Conclusion
Exit Slip – Summarize what you’ve learned about angles and their measures using a bubble map.
Homework – Page 358
Problems 2-22 even