Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half...

179
Ch4.1A – Radian and Degree Measure r

Transcript of Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half...

Page 1: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.1A – Radian and Degree Measure

r

Page 2: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.1A – Radian and Degree Measure

s ~3.14 arcs θ

= half circle r

θ = 1 radian

One radian – the measure of the angle when the arc length = the radius

1 revolution (360˚) = 2π radians (~6.28 arc lengths)

½ revolution ( ) = radians

¼ revolution ( ) = radians

1/3 revolution ( ) = radians

1/8 revolution ( ) = radians

Page 3: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.1A – Radian and Degree Measure

s θ r

θ = 1 radian

One radian – the measure of the angle when the arc length = the radius

1 revolution (360˚) = 2π radians (~6.28 arc lengths)

½ revolution (180˚) = π radians

¼ revolution (90˚) = radians

1/3 revolution (60˚) = radians

1/8 revolution (45˚) = radians

2

3

4

Page 4: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex1) Find the acute angle equivalent to

Ex2) Find the negative angle equivalent to

Ex3) Find the positive angle equivalent to

6

13

4

3

3

2

Page 5: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex4) Find the complement and supplement angles to a)

b)

5

2

5

4

Page 6: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Degree/Radian Conversions

Page 7: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Degree/Radian Conversions

Conversions: Conversion Factor:

rad 00

rad 2360

rad 2

90

rad 6

30

rad 3

60

rad 4

45

rad 180

rad 2

3270

rad 180

rad 180

1

rad

180or

180

rad

Page 8: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex5) Convert:a) 135˚

b) -270˚

c)

d) 2 rad

(Quiz on conv in 2 days) Ch4.1A p318 5-19odd,45-55odd

rad 2

Page 9: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.1A p318 5-19odd,45-55odd Quiz tomorrow! (On conversions)

Page 10: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.1A p318 5-19odd,45-55odd Quiz tomorrow! (On conversions)

Page 11: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.1A p318 5-19odd,45-55odd Quiz tomorrow! (On conversions)

Page 12: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.1A p318 5-19odd,45-55odd Quiz tomorrow! (On conversions)

Page 13: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.1A p318 5-19odd,45-55odd Quiz tomorrow! (On conversions)

Page 14: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.1A p318 5-19odd,45-55odd Quiz tomorrow! (On conversions)

Page 15: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.1B – Arc Length(Quiz tomorrow!)

r

Page 16: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.1B – Arc Length

r

Length of a circular arc: s = r.θ (θ must be in radians)

Ex1) A circle has a radius of 4inches. What is the arc length intercepted by a central angle of 240˚.

Page 17: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Linear speed: distance traveled Angular speed: angle swept out

time time

omega (θ must be in radians)

Ex3) The second hand of a clock is 10.2cm long.Find the speed of the second hand.

t

t

dv

Page 18: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex3) A lawn roller is 30in in diameter and makes 1 revolutionevery 5/6 sec.a) Find the angular speedb) How fast does it move across the lawn?

Page 19: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.1B p319 35-41odd, 52,54,71-81odd,91,95 (Do #35 and #39 in class)(Quiz tomorrow on conversions!)

Page 20: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.1B p319 35-41odd, 52,54,71-81odd,91,95 (Did #35 and #39 in class)(Quiz today on conversions!)

Page 21: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.1B p319 35-41odd, 52,54,71-81odd,91,95 (Did #35 and #39 in class)(Quiz today on conversions!)

Page 22: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.1B p319 35-41odd, 52,54,71-81odd,91,95 (Did #35 and #39 in class)(Quiz today on conversions!)

Page 23: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.1B p319 35-41odd, 52,54,71-81odd,91,95 (Did #35 and #39 in class)(Quiz today on conversions!)

Page 24: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.1B p319 35-41odd, 52,54,71-81odd,91,95 (Did #35 and #39 in class)(Quiz today on conversions!)

Page 25: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.1 Quiz Name___________Convert radians to degrees: A B C D

Convert degrees to radians: A B C D 4. 270˚ 4. 90˚ 4. 180˚ 4. 360˚ 5. 60˚ 5. 30˚ 5. 120˚ 5. 150˚ 6. 210˚ 6. 240˚ 6. 330˚ 6. 300˚

6

7 3.

3

4 3.

3

5 3.

6

11 3.

3

2. 6

2. 6

5 2.

3

2 2.

rad 2

3 1. rad

2 1. rad 2 1. rad .1

Page 26: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.2 – The Unit Circle

x2 + y2 = 1

Ex1) 45˚ = _____ rad

x = _____

y = _____

Page 27: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex2) 60˚ = _____ rad

x = _____

y = _____

Ex3) 30˚ = _____ rad

x = _____

y = _____ Ex5) 90˚ = _____ rad

Ex4) 0˚ = _____ rad x = _____

x = _____ y = _____

y = _____

Page 28: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex6)x = _____ x = _____y = _____ y = _____

x = _____ x = _____y = _____ y = _____

Page 29: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Trig Functions(sine) (cosine) (tangent)

sin t = y cos t = x tan t =

Ex7) Eval 3 trigs for:

a)

b)

c)

d)

Ch4.2A p328 1-39odd (only sin,cos,tan)

x

y

6

t

4

5t

t

2

3t

Page 30: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.2A p328 1-39odd (only sin,cos,tan)

Page 31: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.2A p328 1-39odd (only sin,cos,tan)

Page 32: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.2A p328 1-39odd (only sin,cos,tan)

Page 33: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.2A p328 1-39odd (only sin,cos,tan)

Page 34: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.2B – Trig Functions

sin t = y (cosecant) csc t =

cos t = x (secant) sec t =

tan t = (cotangent) cot t =

Ex8) Eval 6 trigs for:x

y

3

t

y

1

x

1

y

x

Page 35: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

sin t = y csc t =

cos t = x sec t =

tan t = cot t =

Ex9) Eval:

a)

b)

x

y

6

13sin

y

1

x

1

y

x

2

7cos

Page 36: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

x = cos t y = sin t

Domains: (what you put in for t)

Ranges: (what you get out for x or y)

Page 37: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Types of functions:

1. x = cos t is an even function

(So is secant)

2. y = sin t is an odd function

(So is tan, csc, and cot)

Ex10) Use calc:a) sin 76.4˚ (must be in degree mode.)b) cot 1.5 (must be in radian mode.)

Ch4.2B p328 57, 2-38 (eoe)

Page 38: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.2B p328 57, 2-38 (every other even)

Page 39: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.2B p328 57, 2-38 (every other even)

Page 40: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.2B p328 57, 2-38 (every other even)

Page 41: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.2B p328 57, 2-38 (every other even)

Page 42: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.2B p328 57, 2-38 (every other even)

Page 43: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.3 – Right Triangle TrigQuiz tomorrow on this!

SOH-CAH-TOA

sin θ = cos θ = tan θ =

Θhypotenuse

adjacent

opposite

Page 44: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.3 – Right Triangle Trig

SOH-CAH-TOA

sin θ = cos θ = tan θ =

csc θ = sec θ = cot θ =

Ex1) Eval 6 trigs for:

5 4

3

hyp

opp

Θhypotenuse

adjacent

opposite

hyp

adjadj

opp

opp

hyp

opp

hypopp

adj

Θ

Page 45: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.3 – Right Triangle Trig

SOH-CAH-TOA

sin θ = cos θ = tan θ =

csc θ = sec θ = cot θ =

Ex2) Find the value of sin45˚, cos45˚, tan45˚

hyp

opp

Θhypotenuse

adjacent

opposite

hyp

adjadj

opp

opp

hyp

opp

hypopp

adj

45˚

Page 46: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex3) Use the equilateral triangle to find the value of sin60˚, cos60˚, sin30˚, cos30˚

Page 47: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Sine, Cosine, and Tangent of Special Angles

sin30˚ = cos30˚ = tan30˚ =

sin45˚ = cos45˚ = tan45˚ = 1

sin60˚ = cos60˚ = tan60˚ =

HW#8) Find exact values of 6 trigs for:

3

6

Ch4.3A p338 1-22(a,b) Quiz tomorrow – would u like 2 c a sample?

2

1

2

1

2

3

2

3

3

3

3

2

2

2

2

Θ

Page 48: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Sine, Cosine, and Tangent of Special Angles

sin30˚ = cos30˚ = tan30˚ =

sin45˚ = cos45˚ = tan45˚ = 1

sin60˚ = cos60˚ = tan60˚ =

HW#8) Find exact values of 6 trigs for:

3

6

Ch4.3A p338 1-22(a,b) Quiz tomorrow – would u like 2 c a sample?

2

1

2

1

2

3

2

3

3

3

3

2

2

2

2

Θ

Page 49: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Trigonometry & Vector Components

SOHCAHTOA

in

pp

yp

os

dj

yp

an

pp

dj

sinΘ =

cosΘ =

tanΘ =

opphyp

adjhyp

oppadj

Page 50: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.3A p338 1-22(a,b) Quiz today!

Page 51: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.3A p338 1-22(a,b)

Page 52: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.3A p338 1-22(a,b)

Page 53: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.3A p338 1-22(a,b)

Page 54: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.2 Quiz Name___________Find exact values:˚ A B C D

1. sin 30˚ 1. cos 30˚ 1. sin 60˚ 1. cos 60˚2. tan 30˚ 2. tan 60˚ 2. sin 30˚ 2. cos 30˚ 3. cos 60˚ 3. sin 60˚ 3. tan 45˚ 3. tan 30˚4. tan 45˚ 4. cos 45˚ 4. cos 60˚ 4. sin 60˚5. sin 60˚ 5. cos 60˚ 5. tan 30˚ 5. tan 45˚ 6. cos 45˚ 6. tan 45˚ 6. cos 45˚ 6. sin 45˚

Page 55: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.3B – Trig Identities

Reciprocals:

sin θ = cos θ = tan θ =

csc θ = sec θ = cot θ =

Combos:

Pythag: Quiz in 2 days on these identities.

csc

1

sec

1cot

1

sin

1

cos

1tan

1

sin

coscot

cos

sintan

22

22

22

csccot1

sectan1

1cossin

Page 56: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex4) Let θ be acute angle, with sin θ = 0.6, find:a) cos θ b) tan θ

Ex5) Let θ be acute, with tan θ = 3, find:a) cot θ b) sec θ

θ˚

θ˚

Page 57: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex6) Use a calc to eval:a) cos 28˚b) sec 28˚c) sec 5˚40’

Ex7) Find the value of θ in radians and degrees:

a) sin θ = b) cos θ =

b) csc θ = 2

Ex8) Use calc to find θ in:a) degrees for cos θ = 0.3746b) radians for sin θ = 0.3746

Ch4.3B p339 23-31odd,37-40all(a only),47-55all(a only) Quiz in 2 days

2

32

2

Page 58: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.3B p339 23-31odd,37-40all(a only), 47-55all(a only)

Page 59: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.3B p339 23-31odd,37-40all(a only), 47-55all(a only)

Page 60: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.3B p339 23-31odd,37-40all(a only), 47-55all(a only)

Page 61: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.3C – Trig Word Problems (Quiz on ID’s tomorrow!) Ex7) A surveyor is standing 50ft from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5˚. How tall is the tree?

h = ?

θ=71.5˚| --------50ft-----------|

Page 62: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex8) A person is 200yds straight away from a river. The person walks at an angle, going 400yds til he gets to the river’s edge. At what angle did he walk?

200yds 400yds

Page 63: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex9) A 12 meter flagpole casts a shadow 9 meters long. What is the angle of elevation to the sun?

Ch4.3C p33924-32even,68-70all,81,82Quiz tomorrow. Would u like to c an example?

Page 64: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Quiz example:

=

=

csc

1

tan

1

cos

sin

2

2

sec_____1

1cos_____

Page 65: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.3C p33924-32even,68-70all,81,82

Page 66: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.3C p33924-32even,68-70all,81,82

Page 67: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.3C p33924-32even,68-70all,81,82

Page 68: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.
Page 69: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.3C p33924-32even,68-70all,81,82

Page 70: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.3 – Identities Quiz Name__________Reciprocals A B C D

1.

2.

Combinations

3.

Pythag

4.

5.

csc

1

tan

1

cos

sin

2sec___1

sec

1

cot

1

tan

1

cot

1

sin

1

cos

1

cos

sin

sin

cos

sin

cos

1cos___ 2 1___sin 2 ___cot1 2

1___sin 2 1cos___ 2 ___tan1 2 2csc___1

Page 71: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.4A – Trig Functions of Any Angle

(x,y)

Page 72: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.4A – Trig Functions of Any Angle

(x,y) sin θ = cos θ = tan θ = r θ

csc θ = sec θ = cot θ =

Reminder: Ex1) Let -3,4 be a point on the QII QI terminal side of an angle θ. sinθ = sinθ = find sin,cos,tan.cosθ = cosθ =tanθ = tanθ = QIII QIVsinθ = sinθ = cosθ = cosθ =tanθ = tanθ =

r

yr

x

x

y

y

r

x

r

y

x

Page 73: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex2) Given tan θ = . and cos θ > 0. Find sin θ and sec θ.

Ex3) Evaluate sin and tan at

4

5

2

3,,

2,0

Page 74: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Reference Angles- any given angle has an equivalent angle where0 > θ > 90˚ , or 0 > θ > π/2.

Ex4) Find ref angle:a) 300˚b) 2.3c) -135˚

Ex5) Evaluate:a) cos b) tan (-210˚)

c) csc

Quiz in 2 days (Ex?) Ch4.4A p349 2-42eoe (all 6 trigs)

3

4

4

11

Page 75: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Sample Quiz for Ch4.2/4.3

1.

2.

3.

4.

5.

6.

7.

8. )cos(

)300sin(

)270cos(

3

2sin

6

5cos

)135sin(

)135cos(

6

7sin

Page 76: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Lab4.1 – Heights and Lengths (angles measured)

Page 77: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.4A p349 2-42eoe (all 6 trigs)

Page 78: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.
Page 79: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.
Page 80: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.
Page 81: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.4A p349 2-42eoe (all 6 trigs)

Page 82: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.4A p349 2-42eoe (all 6 trigs)

Page 83: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.4A p349 2-42eoe (all 6 trigs)

Page 84: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Lab4.1 – Heights and Lengths (angles measured)

Page 85: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.4B – More Trigs at Any Angle

Ex6) Let θ be an angle in QII, such that sinθ = Find cos θ and tan θ using trig identities.

Ex7) Use a calculator to evaluate:a) cot (410˚)

b) sin (-7)

c) Solve for θ: tan θ = 4.812, where 0 < θ < 2π

Ch4.4B p350 43-81odd (a only) Quiz tomorrow!

3

1

Page 86: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.4B – More Trigs at any Angle

HW#43) Eval sin,cos,tan for:a) 225˚

#63) Find 2 values in degrees and radiansa) sin θ = ½ , where 0 < θ < 2π

Ch4.4B p350 43-81odd (a only) Quiz tomorrow!

Page 87: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.4B p350 43-81odd (a only)

Page 88: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.4B p350 43-81odd (a only)

Page 89: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.4B p350 43-81odd (a only)

Page 90: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.4B p350 43-81odd (a only)

Page 91: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.
Page 92: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.5A – Graphs of Sine and Cosine (Quiz first)Ex1) What are the values of sine and cosine at: θ sin θ cos θ

22

3

2

3

4

6

0

Page 93: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.5A – Graphs of Sine and Cosine (Quiz first)Ex1) What are the values of sine and cosine at: θ sin θ cos θ

Ex2) Graph on a # line: y = sin x

Ex3) Graph on a # line: y = cos x

1 0 2

0 1- 2

3

1- 0

0 1 2

2

1

2

3

3

2

2

2

2

4

2

3

2

1

6

1 0 0

Page 94: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

y = a.sin x y = a.cos x

Amplitude – stretches and shrinks graph vertically

Ex4) Sketch y = 2.sin x

θ 2.sinx

22

3

2

0

Page 95: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

y = a.sin x y = a.cos x

Amplitude – stretches and shrinks graph vertically

Ex5) Sketch y = ½.cos x

θ ½.cos x

22

3

2

0

Page 96: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Period of Sine and Cosine(Normally its 2π)

y = a.sin (bx) y = a.cos (bx)

b determines the period

Ex6) Sketch y = sin(½x) vs y = sin (x) vs y = sin(2x)

1

–1

Ch4.5A p361 1–13odd, 43,45

bperiod

2

4 3 2 2

3

2

Page 97: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.5A p361 1–13odd, 43,45

Page 98: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.5A p361 1–13odd, 43,45

Page 99: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.5A p361 1–13odd, 43,45

Page 100: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.5A p361 1–13odd, 43,45

Page 101: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.5B – Translations

y = a.sin(bx – c) y = a.cos(bx – c)

a = amplitude c = horizontally shifts b determines the period the period

start: bx – c = 0end: bx – c = 2π

Ex7) Sketch y = ½.sin(x – )

bperiod

2

3

2 2

3

2

Page 102: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex8) y = 3.cos(2πx + 1)Use a calc to find its period

Page 103: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex9) Sketch y = 2.cos(2x – π) + 1

Ch4.5B p361 15–21odd,47–53odd (lets do 17 and 47 in class)

Page 104: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.5B p361 15–21odd,47–53odd

Page 105: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.
Page 106: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.
Page 107: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.6A – Graphs of Other FunctionsEx1) Sketch the graph of y = tan x θ tan θ

4

2

2

3

2

4

0

Page 108: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.6A – Graphs of Other FunctionsEx1) Sketch the graph of y = tan x θ tan θ

To find asymptotes for tangent:0 2

undefined 2

3

0

undefined 2

1 4

0 0

2

2

cbx

cbx2

2

Page 109: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex2) Sketch the graph of y = tan To find asymptotes θ tan θ for tangent:

4

2

2

3

2

4

0

2

x

2

2

cbx

cbx

Page 110: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex3) Sketch the graph of y = –3tan2x To find asymptotes θ tan θ for tangent:

4

2

2

3

2

4

0

2

2

cbx

cbx

Page 111: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex4) Sketch the graph of y = cot x θ cot θ

4

2

2

3

2

4

0

sin

coscot

Page 112: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex5) Sketch the graph of y = 2cot To find asymptotes θ cot θ for cotangent:

Ch4.6A p372 9-12,21,22,24

4

2

2

3

2

4

0

cbx

cbx 0

3

x

Page 113: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.6A p372 9-12,21,22,24

Page 114: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.
Page 115: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.6B – Graphs of Other FunctionsEx6) Sketch the graph of y = csc x θ sin x csc x

To find asymptotes for cosecant:

anywhere sin x = 0

2

2

3

2

4

0

Page 116: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex7) Sketch the graph of y = sec x θ cos x sec x

To find asymptotes for secant:

anywhere cos x = 0

2

2

3

2

4

0

Page 117: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

HW#13) Sketch the graph of y = -½sec x

Page 118: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

HW#19) Sketch the graph of y = csc

Ch4.6B p372 13-20all,23,26

2

x

Page 119: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.6B p372 13-20all,23,26

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Page 121: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.7 – Inverse Trig Functions

Ex1) Graph y = sin x

1

-1 2

2

3

2

2- -

Page 122: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.7 – Inverse Trig Functions

Ex1) Graph y = sin x

1

-1

The inverse function of sin x is called sin-1 x or arcsin x - its domain is [-1,1], and its range is .

Graph arcsin x

2 2

3

2

2- -

2,

2

Page 123: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

If sin θ = then sine is taking an angle and giving us the ratio of the side opposite to the hypotenuse.

If sin-1 = θ then inverse sine is taking the ratio of the sidesand giving us the angle

Ex2) a) Find the exact value of arcsin(- ½)

b) Find the exact value of arcsin( )

c) Find the exact value of arcsin(2)

r

y

r

y

2

3

Page 124: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex3) Graph y = cos x

1

-1 Graph y = arccos x

Ex4) Find the exact value of arccos(½)

Find the exact value of arccos ( )

2 2

3

2

2- -

2

3

Page 125: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex5) Graph y = tan x

1

-1

Graph y = tan-1 x

Ex6) Find the approx value of tan-1 (.7042)

Ch4.7A p383 7-19odd (a,b),23,25

2 2

3

2

2- -

Page 126: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.7A p383 7-19odd (a,b),23,25

Page 127: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.7A p383 7-19odd (a,b),23,25

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Page 129: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.7B – Inverse Trig Functions cont

Ex6) Graph y = sin

1

-1

Graph y = sin-1

2 2

3

2

2- -

2

x

2

x

Page 130: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Inverse propertiessin(arcsin x) = x arcsin(sin x) = xcos(arccos x) = x arccos(cos x) = xtan(arctan x) = x arctan(tan x) = x

Ex7) Find the exact value of tan(arctan(-5))

Find the exact value of arcsin(sin )3

5

Page 131: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.7B p383 8-20even (a only),27-41odd

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Page 133: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.
Page 134: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.7C – Inverse Trig Functions cont

Ex7) Find the exact value of a. tan(arccos( )

b. cos(arcsin( )

5

2

5

3

Page 135: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex8) Write each as an algebraic expression: a. sin(arccos(3x) 0 < x < 1/3

b. cot(arccos(3x) 0 < x < 1/3

Page 136: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.7C p385 34-42even,43-51odd,71,75

Page 137: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.7C p385 34-42even,43-51odd,71,75

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Page 139: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.
Page 140: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

34.2˚

c

b = 19.4

a

Ch4.8A – Applications

Ex1) Solve the triangle: B

CA

Page 141: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

72˚

c =1

10ft

h = ?

Ex2) The maximum angle for a ladder is 72˚.If a fire dept’s longest ladder is 110ft, what is the max rescue height?

Page 142: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

35˚

h = ?

Ex3) At a point 200ft from the base of a building,the angle of elevation to the bottom of a smoke stack is 35˚.The angle of elevation to the top of the smoke stack is 53˚.Find the height of the smoke stack.

53˚

Page 143: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

3.9m

20m

Ex4) Find the angle of depression to the bottom of a pool.

Ch4.8A p394 1-11odd,18

1.3m

Page 144: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.8A p394 1-11odd,18

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Page 146: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.8B – Applications

Ex5) A ship leaves a port with a heading of N54˚W traveling at 20mph.Ship 2 leaves port at the same tjme with a heading N36˚E traveling at 30mph. After 2 hours how far apart are they?

Page 147: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

HW#35) An observer in a lighthouse 350ft above sea level observes2 ships directly offshore. The angle of depression to the ships are 4˚ and 6.5˚. How far apart are the ships?

Ch4.8B p395 17-37odd

4˚ 6.5˚

Page 148: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Lab4.2 – Finding Angles

Page 149: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.8B p395 17-37odd

Page 150: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.8B p395 17-37odd

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Page 152: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.8B p395 17-37odd

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Page 154: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.
Page 155: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.
Page 156: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.
Page 157: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.
Page 158: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.8C – Simple Harmonic Motion (SHM)

10cm

0cm

-10cm

Page 159: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

10cm

0cm

-10cm

d = a.sinωt or d = a.cosωt

|a| = amplitude

= period

= frequency

2

2

Page 160: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex6) a) Write an equation for the SHM of a ball attached to a spring, that is pushed up 10cm, and oscillates with a period of 4sec.

b) Find the frequency.

Page 161: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ex7) Given a spring in SHM described by: d = 6cos[ t]find:a) Periodb) Frequencyc) Where is the ball located when t = 4?d) Find 2 times where d = 0.

4

3

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Lab4.3 – Simple Harmonic Motion

Go over HW quickly

HW: Finish lab questions+

Ch4 Rev#1 p401 1-53eoeCh4 Rev#2p402 61,65,69,73,75,77, 87,93,97,99,103,104,105,107

Page 163: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4.8C p398 20,49-56all,58 (let’s do 52 and 56 in class #20 on next slide)

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Page 165: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.
Page 166: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4 Rev#1 p401 1-53eoe Ch4 Rev#2p402 61,65,69,73,75,77, 87,93,97,99,103,104,105,107

Page 167: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4 Rev#1 p401 1-53eoe

Page 168: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4 Rev#1 p401 1-53eoe

Page 169: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4 Rev#1 p401 1-53eoe

Page 170: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4 Rev#2p402 61,65,69,73,75,77,87,93,97,99,103,104,105,107

Page 171: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

Ch4 Rev#2p402 61,65,69,73,75,77,87,93,97,99,103,104,105,107

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Page 173: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.
Page 174: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.
Page 175: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.
Page 176: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.
Page 177: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.
Page 178: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.
Page 179: Ch4.1A – Radian and Degree Measure r. Ch4.1A – Radian and Degree Measure s ~3.14 arcs θ = half circle r θ = 1 radian One radian – the measure of the angle.

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