Unit 1, Lesson 1Angles and their Measures

What is an angle?

Two rays with the same Endpoint.

Make a table

Term Definition Picture

Trigonometry vocabularyInitial side – start side of angle

Terminal side – end side of angle

Standard position – An angle whose initial side is on the positive x-axis

Coterminal angle – Angles that have the same terminal side

Positive angles

• Initial side on the positive x-axis and rotate counterclockwise

Negative angles

• Initial side on the positive x-axis and rotate clockwise

Quadrants

Quadrant III

Quadrant IQuadrant II

Quadrant IV

Angles of the axes

0 00 ,360

0 090 , 270

0 0180 , 180

0 0270 , 90

• Variables you will see for angle measures theta

alpha

beta

Examples

• What quadrant is the terminal side of the angle in? (Make a sketch of the angle) 332

156

1000

240

o

o

o

o

Examples - Coterminal angles

• You can add or subtract multiples of 360 or -360 to find coterminal angles

• Find 2 coterminal angles (one positive and one negative) for 35o

• Find 2 coterminal angles (one positive and one negative) for -23o

• Find 2 coterminal angles (one positive and one negative) for 740o

Decimal degree vs. degree/minute/second

• Sometimes angles are not whole numbers• When this happens, they take 2 forms

– Degree decimal - 76.5o

– Degree/minute/second (DMS) 53o 18’ 23”

• To change from DMS to decimals, divide the minutes by 60 and the seconds by 3600 and add all together

• To change from decimal to DMS, multiply decimal by 60 – this is the minutes, then multiply resulting decimal by 3600, this is the seconds

Examples - DMS

• Convert to decimal• 35o 18’ 27”

• -142o 54’ 32”

• Convert to DMS• -23.873o

• 740.5o

Radians

Examples

• Change from degrees to radians

• Change from radians to degrees

36

250

360

o

o

o

4

3

16

Coterminal Examples using radians

• Add or subtract 2π to angle

4

3

16

Complementary/Supplementary

• Complementary angles add up to 90o or

• Supplementary angles add up to 180o or

2

Examples

• What angle is complementary to 36o ?

• What angle is supplementary to 36o ?

• What angle is complementary to ?

• What angle is supplementary to ?

3

3

Finding arc lengths• S=rθ• S is arc length• r = radius• θ = central angle, must be in radians

Examples• Find the length of the arc with radius 20 in

and central angle of π/4

• Find the length of an arc with radius 5 m and central angle of 180o

• Find the measure of the central angle is arc length is 6 in and radius is 18 in

• θ = central angle