6.1: Angles and their measure

January 5, 2008

Objectives

• Learn basic concepts about angles• Apply degree measure to problems• Apply radian measure to problems• Calculate arc length• Calculate the area of a sector

What is an angle?

• An angle is formed by rotating a ray around its end point.

• Important terms:– Initial side: starting

position of the ray– Terminal side: the final

position of the ray– Positive measure: ray is

rotated counterclockwise– Negative measure: ray is

rotated clockwise

Degree measure

• One complete rotation is 360°.

• 90° is a right angle.• 180° is a straight

angle.• Symbols used to

denote angles:– Alpha - α– Beta - β– Theta - θ

Important angle terms

• Complementary angles add to be 90°.

• Supplementary angles add to be 180°.

• Acute angles 0<θ<90.• Obtuse angles

90<θ<180.• Coterminal angles:

angles with the same terminal side.

Radian measure

• The circumference of a circle is 2π.

• Therefore, one rotation of ray is 2π radians.

• To convert from degrees to radians..Multiply degrees by π/180°

• To convert from radians to degrees..Multiply radians by 180°/π

• 2π = 360°• π = 180°• π/2 = 90°• π/3 = 60°• π/4 = 45°• π/6 = 30°

Try these

Degree to radian

120°

150°

200°

320°

Radian to degree

2π/5

3π/4

7π/5

6π/5

Arc length

• Arc length

s= rθ• θ must be in radian

measure.

Try it

A circle has a radius of 4. Find the length of an arc intercepted by a central angle of 60°.

Try this one

A circle has a radius of 12. The arc length of a certain angle is 4. Find the central angle.

Area of a sector

• Area of a sector

A= (1/2)r2 θ• θ must be in radian

measure.

Try it

A circle has a radius of 5. Find the area of the sector if the central angle is 75°.

Your assignment

1,2 – sketching angles

21-26 – complementary and supplementary

35-38 – find the central angle

43, 44 – converting from degrees to radians

47-52 – find the missing value (arc length)

65-68 – area of a sector