m12pc45

SECT. 4.5 APPLICATIONS OFTRIGONOMETRY

REVIEW: SOLVING

“Solving” means finding a value for the variable, so thatboth sides of an equation will be equal

There are several ways to solve a system of equations:

Algebraically:

Chapter 5 – Solving Trigonometric Solutions

Graphically:

Make Y1 equal to the trig. Function

Make Y2 equal to the value you want the trig. Function to beequal to

Then find the points of intersection(s)

Points where the graph intersect  the x-coordinates are thesolutions

SOLVING TRIGONOMETRIC FUNCTIONS WITH TI-83

Make the left side of the equation y1

Make the right side of the equation y2

Solve the system by findingthe points of intersection

Press:

2nd

TRACE

#5: Intersect

1st Curve?

ENTER

2nd Curve?

ENTER

Guess?

ENTER

Repeat this process untilyou find all the points ofintersection

I) APPLICATIONS OF TRIGONOMETRIC FUNCTIONS IN THEREAL WORLD:

Mechanical Engineering

Circular or Periodic Motions

Bicycles, Ferris Wheels,

Simple Harmonic Motions (Springs)

Ecological Studies

Ocean Tides (High and Low tides)

Ocean Depths

Astronomy

Solar System, Hayley’s Comet

Physics

Waves: Sound waves, Electromagnetic waves

Light

II) OCEAN TIDES

Earth's ocean surface will rise and fall periodically

High tide occurs when water level peaks

Low tide occurs when ocean levels is at it’s minimum

When water levels begin to change (Slack point/ Turning)

Caused by the gravitational pulls from the Moon and the Sunacting on the oceans (wikipedia)

As the moon rotates around the earth, different parts of themoon will generate different amounts of gravitatonal pull onthe ocean surface

Important for coastal navigation

http://en.wikipedia.org/wiki/Tidal_force

High tide at

Bay of Fundy

Low tide at

Bay of Fundy

High tide occurs at 3:30am

with ocean depths at 10.9m

Low tide occurs 8.5 hours laterwith ocean levels at 3.9m

Ocean levels will fluctuate

up and down over several hours

Ex: If high tide (10.6m) occur at 3:30am andlow tide (3.9m) is 8.5 hours later, write anequation for the depth of the water at anytime, t hours.

Ex: If high tide (10.6m) occur at 3:30am and low tide(3.9m) is 8.5 hours later, write an equation for thedepth of the water at any time, t hours.

High Tide

Low Tide

High Tide

III) FERRIS WHEEL PROBLEM:

A Ferris Wheel at a park has riders get on at the bottom of thewheel, which is 11m above the ground. The highest point of theride is 55m above ground. The ride takes 120s to complete onerevolution. Derive a formula in the form of :for the height of a rider.

100

120

The height of the rider v.s. time creates a sinusoidal wave

To find the formula for the equation, use the graph to obtain theamplitude, period, vertical & horizontal shifts

100

120

No horizontal shift, the graph begins at t = 0.

However, there’s a vertical reflection. (UpsideDown Cosine Function) Graph begins at thebottom

Plug the values into the trigonometricfunction

Q1: Find the height of the rider 30seconds after the ride begins:

Q2: Within 1 complete revolution, what is the total amount oftime a rider on the ferris wheel will spend above 40m?

100

120

Find the intersection pointsbetween the two functions

The time above 40m will be

the horizontal differencebetween the two points

The height of the riderafter 30s will be 33metersabove the ground

IV) EARTH AND SUN IN THE SOLAR SYSTEM

The earth circulates in an elliptical orbit around the sun

Perogee (Perihelion) point in orbit where the earth-sundistance is at its minimum (early January - Winter)

Distance = 147.5 million km (2008)

Apogee (Aphelion) point in orbit where the earth-sundistance is at its maximum (early July - Summer)

Distance = 152.6 million km (2008)

Earth’s orbit is stabilizing, but changes slightly every year

http://en.wikipedia.org/wiki/Apsis

Note: Seasons depend on earth’s axis of orbit, not on earth-sun distance

Perihelion: distancebetween the earth andsun is shortest at this point

D = 147.5million km

Aphelion: Distance between the earth and sun is longest at this point.

D=152.6 million km

http://www.spaceweather.com/glossary/aphelion.html

Ex: In 2008, perihelion is on January 8 and aphelion is on July 4.Express the distance, d million km, from Earth to sun as asinusoidal function of the number of days of the year.

b) Use the function to find the earth/sun distance on Dec 2, 2008

Perihelion

Aphelion

HOMEWORK:

P276 #1 – 11