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Trigonometric ModelsChapter9

• Trigonometric Functions, Models, and Regressions (Lecture 21, Wed. 3/28/07)

• Derivatives of Trigonometric Functions and Applications (Lecture 22, Mon. 4/2/07)

• Integrals of Trigonometric Functions and Applications (Lecture 23, Wed. 4/4/07)

Lecture 21

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• Trigonometric Functions, Models, and Regressions

Examples:

• Sunspot Activity, Piston in a Cylinder

• Tides, AC Current and AC Voltage

• Average Daily Air Temperature

• Seasonal Fluctuations in Business Demand

• Hair Cut/Growth, Fluffy Paces on the Ledge, Sound Waves, …etc.

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Calculating sin(Calculating sin(), cos(), cos(), and tan(), and tan() ) Triangle method: SOH, CAH, TOATriangle method: SOH, CAH, TOA

hypotenuse (le

ngth = h)

adjacent side (length = x)

op

po

site. s

ide

(len

gth

= y)

• ““SOH” SOH” ssin(in() = () = (oopposite side)/pposite side)/hhypotenuseypotenuse

• ““CAH” CAH” ccos(os() = () = (aadjacent side)/djacent side)/hhypotenuseypotenuse

• ““TOA” TOA” ttan(an() = () = (oopposite side)/(pposite side)/(aadjacent side)djacent side)

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The Sine FunctionThe Sine FunctionThe sine of a real number t is the y–coordinate (height) of the point P in the following diagram, where |t| is the length of the arc.

x

y

Psin t

1 unit

1

1

–1

–1 |t|

function. An odd

tt sinsin

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The Sine Function

sin(0) 0

sin 12

sin 0

3sin 1

2

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0.5 1 1.5

Highlight those sections of the circle where sin(t) >0

sin(t) > 0sin(t) > 0

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The Sine Function

2

2

3

2

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The General Sine Function

( ) sinf x A x C

A is the amplitude (peak height above baseline)

C is the vertical offset (height of baseline)

P is the period (wavelength)

is the angular frequency

is the phase shift

2 /P

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Sine Function

( ) 1.5sin 0.5 1.3 1.8f x x

1.5 is the amplitude

1.8 is the vertical offset

is the period

0.5 is the angular frequency

2.6 is the phase shift

2 / .5 4P

Example:

1.5sin 0.5( 2.6) 1.8x

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Sine Function

( ) 1.5sin 0.5 1.3 1.8f x x Example:

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Basepoint

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The Cosine FunctionThe Cosine FunctionThe cosine of a real number t is the x–coordinate (length) of the point P in the following diagram, where |t| is the length of the arc.

x

y

P

cos t

1 unit

1

1

–1

–1 |t|

function. An even

tt coscos

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The Cosine Function

cos(0) 1

cos 02

cos 1

3cos 0

2

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0.5 1 1.5cos(t) > 0cos(t) > 0

Highlight those sections where cos(t) > 0

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The Cosine Function

2

2

3

2

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The General Cosine Function

( ) cosf x A x C

A is the amplitude (peak height above baseline)

C is the vertical offset (height of baseline)

P is the period (wavelength)

is the angular frequency

is the phase shift

2 /P

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Fundamental Trigonometric Identities (Relationships Between Cosine and Sine)

2 2sin cos 1t t

cos sin / 2

sin cos / 2

t t

t t

cos sin / 2

sin cos / 2

t t

t t

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0.5 1 1.5

-1

0

1

-9 -6 -3 0 3 6 9

sin(t)

cos(t)

Alternative:

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Other Trigonometric Functions

Tangent:sin

tancos

xx

x

Cotangent:

Secant:

Cosecant:

cos 1cot

sin tan

xx

x x

1sec

cosx

x

1csc

sinx

x

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Other Trignometric FunctionsOther Trignometric FunctionsThe Tangent and Cotangent FunctionsThe Tangent and Cotangent Functions

y= tan(x)

tan(tan(xx) = sin() = sin(xx)/cos()/cos(xx))

y = cot(x)

cot(x) = cos(x)/sin(x) = 1/tan(x)

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Trigonometric RegressionUse data that suggests a sine (or cosine) curve and perform a regression to find the best-fit generalized sine (or cosine) curve.

time (months) Line Fit Plot

200

250

300

0 100 200 300 400

time (months since Jan. 1961)

Ozo

ne

(D

ob

so

n

un

its

)

Ozone (Dobson units) Predicted Ozone (Dobson units)

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Example: Example: Cash Flows into Stock FundsCash Flows into Stock Funds

The annual cash flow into stock funds (measured as a percentage of total assets) has fluctuated in cycles of approximately 40 years since 1995, when it was at a high point. The highs were roughly +15% of total assets, whereas the lows were roughly 10% of total assets.

a. Model this cash flow with a cosine function of the time t in years, with t = 0 representing 1995.

b. Convert the answer in part (a) to a sine function model.

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Example: Example: Cash Flows into Stock FundsCash Flows into Stock Funds

arradians/ye 157040

22 is,that

years, 40 period, Given the

.

P

P

52 shift, Vertical

% 5122

1015 Amplitude,

%.

.

C

A

Solution:

1995. since yearsin timeis and ,in flowcash theis where,

522

1570512 or,

521570512 Thus,

ttC

ttC

ttC

%

..sin.

..cos.

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AnotherAnother Example Example of Periodic Systems of Periodic SystemsSunspot CountsSunspot Counts

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Section 9.1 Problem #39 (Waner pp. 552Section 9.1 Problem #39 (Waner pp. 552553)553)

Sunspot Activity The activity of the Sun (sunspots, solar flares, and coronal mass ejection) fluctuates in cycles which can be modeled by

8584316020757 ...sin. ttN

a. What is the period of sunspot activity according to this model?

b. What is the maximum number of sunspots observed? What is the minimum number?

c. When to the nearest year, is sunspot activity next expected to reach a high point?

Where t is the number of years since January 1, 1997, and N(t) is the number of sunspots observed at time t.

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Section 9.1 Problem #39 (Waner pp. 552Section 9.1 Problem #39 (Waner pp. 552553)553)

years 4106020

22 thus,

ar,radians/ye 6020 asgiven is Since

..

.

P

sunspots 118580757

sunspots 51168581757

...

...

min

max

N

N

Solution:

2011). Junein (or, years 5146020

25431for Solve

2

54316020 occur when willNext

..

/.

max

t

.t.N