Angles and Their Measures

Chapter 4, Sections 1 & 3

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.

C A è

Terminal Side Initial Side

B

Vertex

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.

Standard Position – Positive Angles

y

 is

positive

Terminal Side 

x

Vertex Initial Side Positive angles rotate

counterclockwise

.

Standard Position – Negative Angles

y

Vertex Initial Side

x

 Terminal Side  is

negative

Negative angles rotate

clockwise

.

Measuring Angles  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

Angle Conversions – Degrees to DMS 1.

2.

Keep the whole number – this is the degree part (ᵒ) Multiply the decimal part by 60 – this is the minutes part (‘) 3.

Multiply the remaining decimal part by 60 – this is the seconds part (“)

Angle Conversions – DMS to Degrees 1.

Keep the degree part – this is the whole number.

2.

Divide the minutes part by 60, divide the seconds part by 3600, then add those two numbers to each other – this is the decimal.

Angle Conversions – Degrees to Radians 1.

2.

Divide the degrees by 180 Multiply by π **leave in fraction form** **if the angle is in DMS form, convert back to degrees first**

Angle Conversions – Radians to Degrees  Divide by π  Multiply by 180

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 360 º  The radian measure of an angle has been increased/decreased by a multiple of 2 π

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

Special Angles – Reference Angles

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

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