Table of Contents1. Angles and their Measures

Angles and their Measures

Essential question – What is the vocabulary we will need for

trigonometry?

Make a table

• TermDefinition 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

Positive angles

• An angle in standard position that rotates counterclockwise

Negative angles

• An angle in standard position that rotates clockwise

Coterminal Angles

• Angles that have the same terminal side

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

Radians

• Angle measures can also be expressed in radians• A radian is the ratio of the length of an arc to its

radius• Radians are expressed in terms of• = 180o

• To change from degrees to radians, multiply by and reduce.

• To change from radians to degrees, multiply by

180

180

Radians continued

• Radians can take 2 forms – an exact answer and an approximate decimal answer

• The exact answer has a in it and it is the usual way to see radians

• To find an exact answer with your calculator, do not put the in the calculator, only write it in the answer

• However, radians can also be written as a decimal without the

Angles of the axes

0,2

2

3

2

Examples

• Change from degrees to radians

• Change from radians to degrees

36

250

360

o

o

o

4

3

16

Coterminal angles

• You add or subtract multiples of 360o (or 2π) 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

Examples for radians

• Find a positive and negative coterminal angle

4

3

16

What quadrant is it in?• To find out what quadrant an angle is in

– Make a negative angle positive by adding 360o or 2π (may need to do multiple times)

– If angle is bigger than 360o or 2π, make it smaller by subtracting 360o or 2π (may need to do multiple times)

– Figure out what quadrant it is in based on angles of axes (from yesterday)

– If the question asks you to sketch the angle,• draw the terminal side in the right quadrant• go in either positive or negative direction based on original

problem• if you have added or subtracted 360o or 2π, you need to go

around multiple times.

What quadrant is it in (and sketch)?

332

156

1000

240

o

o

o

o

What quadrant is it in? (radians)

• Follow steps to make small positive angle• Put fraction in calculator (without the π)• If answer is < 0.5, it is in 1st quadrant• If answer is between 0.5 and 1, it is in 2nd quadrant• If answer is between 1 and 1.5, it is in 3rd quadrant• If answer is between 1.5 and 2, it is in 4th quadrant

Examples – which quadrant? (radians) (and sketch)

7

5

3

16

34

5

Reference Angles• A reference angle is the acute angle that

an angle makes with the x-axis•

Finding Reference Angles

• Follow steps to make small positive angle• Find out which quadrant it is in• In the 1st quadrant, the reference angle is the

SAME as the angle itself• In the 2nd quadrant subtract the angle from 180o or

π• In the 3rd quadrant subtract 180o or π from the

angle• In the 4th quadrant subtract the angle from 360o or

2π

Examples

• Find the reference angle for the following angles.• 37o

• 7π/4• -2π/3• -190o

• 17π/7• 820o

Assessment

• 321– Write 3 new things you learned– Write 2 vocabulary words with their meaning– Write 1 thing you don’t understand