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. A è B C Terminal Side Initial Side 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 x y Terminal Side Initial Side Vertex   is positive Positive angles rotate counterclockwise.

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

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

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 (“)

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.

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**

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

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 90º 180º 270º 360º

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