Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent...

Radians and AnglesWelcome to Trigonometry!!

StarringThe Coterminal Angles

Sine

Cosine

Tangent

Cosecant

Cotangent

Secant

Angles

Radian

Degree

Degree MeasureOver 2500 years ago, the Babylonians used a number system based on 60

The number system we use today is based on 10

However we still use the Babylonian idea to measure certain things such as time and angles. That is why there are 60 minutes in an hour and 60 seconds in a minute.

The Babylonians divided a circle into 360 equally spaced units which we call degrees.

In the DMS (degree minute second) system of angular measure, each degree is subdivided into 60 minutes (denoted by ‘ ) and each minute is subdivided into 60 seconds (denoted by “)

Since there are 60 ‘ in 1 degree we can convert degrees to minutes by multiplying by the conversion ratio

0

'

1

60

Convert 34.80 to DMS

We need to convert the fractional part to minutes

'48608.

'00 48348.34

Convert 112.420 to DMS

Convert the fractional part

'2.256042. Convert the fractional part of the minutes into seconds

''12602. '''00 122511242.112

Convert 42024’36’’ to degrees

This is the reverse of the last example. Instead if multiplying by 60, we need to divide by 60

000

0'''0 41.426060

36

60

2442362442

Radian Measure

1

The circumference of a circle is 2πrIn a unit circle, r is 1, therefore the circumference is 2π

A radian is an angle measure given in terms of π. In trigonometry angles are measured exclusively in radians!

Radian Measure

1

Since the circumference of a circle is 2π radians, 2π radians is equivalent to 360 degrees

Radian Measure

1

Half of a revolution (1800) is equivalent to

22

1radians

Radian Measure

1

One fourth of a revolution (900) is equivalent to

24

22

4

1 radians

Since there are 2π radians per 3600, we can come up with the conversion ratio of

360

2

180

Which reduces to

radians

degrees

radians

degrees

To convert degrees to radians multiply by

180

radians

degrees

To convert radians to degrees multiply by

180

radians

degrees

To convert 900 to radians we can multiply

00

18090

radians

2

2900

radians

We also know that 900 is ¼ of 2π

24

22

4

1 radians

Arc length formula

θ

r

If θ (theta) is a central angle in a circle of radius r, and if θ is measured in radians, then the length s of the intercepted arc is given by

s

rs THIS FORMULA ONLY WORKS WHEN THE ANGLE MEASURE IN IS RADIANS!!!

Angle- formed by rotating a ray about its endpoint (vertex)

Initial Side Starting position

Terminal Side Ending position

Standard PositionInitial side on positive x-axis and the vertex is on the origin

Angle describes the amount and direction of rotation

120° –210°

Positive Angle- rotates counter-clockwise (CCW)

Negative Angle- rotates clockwise (CW)

Coterminal Angles

• Angles with the same initial side and same terminal side, but have different rotations, are called coterminal angles.

• 50° and 410° are coterminal angles. Their measures differ by a multiple of 360.

Q: Can we ever rotate the initial side counterclockwise more than one revolution?

Answer – YES!

EXITBACK NEXT

Note: Complete Revolutions

Rotating the initial side counter-clockwise

1 rev., 2 revs., 3revs., . . .

generates the angles which measure

360, 720, 1080, . . .

EXITBACK NEXT

Picture

EXITBACK NEXT

ANGLES 360, 720, & 1080 ARE ALL COTERMINAL

ANGLES!

What if we start at 30 and now rotate our terminal side counter-clockwise 1 rev., 2 revs., or 3 revs.

EXITBACK NEXT

Coterminal Angles: Two angles with the same initial and terminal sides

Find a positive coterminal angle to 20º 38036020

34036020

Find 2 coterminal angles to 4

15

4

8

4

15

24

15

4

8

4

15

24

15

4

23

4

8

4

7

Find a negative coterminal angle to 20º

4

Warm Up

• Convert to Degrees minutes, seconds

• Convert to Radians:

225 72

735.15

Now, you try…

Find two coterminal angles (+ & -) to 3

2

What did you find?

3

8,

3

4

These are just two possible answers. Remember…there are more!

Complementary Angles: Two angles whose sum is 90

Supplementary Angles: Two angles whose sum is 180

6

62

36

2

66

3

3

2

3

233

2

3

3

To convert from degrees radians, multiply by

To convert from radians degrees, multiply by

180

180

Convert to radians:

180

135

4

3

180

80

9

4

To convert from degrees radians, multiply by

To convert from radians degrees, multiply by

180

180

Convert to degrees:

180

3

8 480

180

6

5 150

So, you think you got it now?

Express 50.525 in degrees, minutes, seconds

50º + .525(60) 50º + 31.5

50º + 31 + .5(60)

50 degrees, 31 minutes, 30 seconds

CW/HW

• Page 280-281 (1, 3, 5-8, 11-14, 30-33)