More Trigonometry!!Section 4-2

ReviewAngles

Standard Position

Coterminal Angles

Reference Angles

Converting from Degrees –

degrees, minutes, seconds (DMS)

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

Initial Side Starting position

Terminal Side Ending position

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

An angle describes the amount and direction of rotation120° –210°

Positive Angle- rotates counter-clockwise (CCW)

Negative Angle- rotates clockwise (CW)

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

Find a positive coterminal angle to 20º 38036020 34036020Find a negative coterminal angle to 20º

Types of questions you will be asked:

Identify a) ALL angles coterminal with 45º, then b) find one positive coterminal angle and one negative coterminal angle.

a) 45º + 360k (where k is any given integer).

b) Some possible answers are 405º, 765º, - 315º, - 675º

Decimal Degrees (DD)

• Decimal degrees are similar to degrees/ minutes/seconds (DMS) except that minutes and seconds are expressed as decimal values.

• Decimal degrees make digital storage of coordinates easier and computations faster.

60.34444 instead of 60°20'40"

1 degree = 60 minutes

1° = 60

1 minute = 60 seconds

1 = 60

So … 1 degree = _________seconds

3600

Express 365010as decimal degrees (DD)

To complete the calculation, remember that …

Converting from DMS to DD

THEREFORE …

Try this: Converting DMS to DD

20 minutes.= 0.33333 (20/60)40 seconds = 0.01111 (40/3600)

Add up the degrees to get an answer:60º + 0.33333 + 0.01111=60.34444 DD

60º20'40"degrees

minutes

seconds

Express 50.525 in degrees, minutes, seconds

50º + .525(60)

50º + 31.5

50º + 31 + .5(60) 50 degrees, 31 minutes, 30 seconds

Converting from DD to DMS

To reverse the process, we multiply by 60 instead.

Homework

Page 238 # 2 - 16 evens

So, what exactly is a RADIAN?

Many math problems are more easily handled when degrees are converted to RADIANS.

For a visual depiction of a radian, let’s look at a circle.

θ1 radian2

3

4

56

a little extrar

So, how many radians are there in a given circle?

What’s the connection between degrees and radians?

360 2 r

3602

r 180 57.3

Definition: a radian is an arc length of one radius

We can use the two ratios to convert between radians and degrees.

Example: Change 330˚ to radians:

Example: Convert radians to degree measure.

180180

or

330180

116

23

2 1803

120

In most cases, radians are left in terms of π

Two formulas to know:

1. Arc Length of a circle: S = rθ (θ in radians)

Example: Given a central angle of 128 degrees, find the length of the intercepted arc in a circle of radius 5 centimeters. Round to nearest tenth.

S = rθ

2. Area of a sector (slice of pie): A = ½ r2θ (θ in radians)

Example: Find the area of a sector of the central angle measures radians and the radius of the circle is 16 inches. Round to nearest tenth.

5 128180

11.2 cm

A = ½ r2θ 2 21 516 335.12 6

in

Linear & Angular VelocityThings that turn have both a linear velocity

and an angular velocity.

Things that Turn - Examplestire on a car or bike

buckets on a waterwheel

teeth on a gear

can on a kitchen cabinet lazy susan

propeller on an airplane

horse on a Merry-Go-Round

fins on a fan or a windmillearth on its axis

Linear & Angular Velocity - Examples

film on a projector or tape on a videotape

turntable in a microwave oven

blade on a lawnmower

Earth around the sun

rope around a pulleyseat on a Ferris wheel

a record on an old record playerdrum/barrel in a clothes dryer

Things that Turn - Examples

lock on your lockerhands on a clockroller brush on a vacuum cleanertops & gyroscopes & dradle motor crankshaftblades in a blender roller skate wheelsCarnival rides: tilt-a-whirl, scrambler, etc.weather vane washing machine agitator

Angular Velocity

Angular Velocity (ω): the speedat which an angle opens. t

Definition:

Remember: θ is in radians.

Ex. 6 rev/min, 360°/day, 2π rad/hour

Angular VelocityExample: determine the angular velocity if 7.3 revolutions are completed in 9

seconds. Round to nearest tenth.

1 revolution is 2π radians … so we’re talking about…

Let’s use the formula:

7.3 2 14.6 radians

t

14.6 9.2 / sec9sec

rad

Angular Velocity

EXAMPLE 2: A carousel makes 2 5/8 rotations per minute. Determine the angular velocity of a rider on the carousel in radians per second .

582 2.625revolutions

2.625 1min 2 0.2751min 60sec sec

revs radians radiansrevolution

Linear Velocity

v rt

Linear Velocity: the speed with whichAn object revolves a fixed distance from a central point.

Definition:

Ex. 55 mph, 6 ft/sec, 27 cm/min, 4.5 m/sec

If you already know the angular velocity, then …

r

Linear VelocityIn the carousel scenario, one of the animals is 20 feet from the center. What is its linear

velocity?

SolutionThe cable moves at a fixed speed … a linear velocity.

rt .27520

secradians

5.5 ft/sec