Trig Applications

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Trig Applications

• Or
• This is When We Will Use This

2
Problem Types

• Right Triangles
• Non-Right Triangles
• Law of Sines
• Law of Cosines
• Area of Non-Right Triangles using Right Triangles
• Sine Curves and Curve Fitting

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Right Triangles

• Types
• Given one angle and one side
• River Problem
• Angle of elevation/declination Problem
• Two Sightings Problem
• Given two sides
• Inclination Problem
• Bearing of an Airplane Problem

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Right Triangles Type 1One side, One angle
To solve problems of this type, use the angle and
the side in a trig function basic definition and
solve for the other side. Tangent for
opposite-adjacent relationships Sine for
opposite-hypotenuse relationships Cosine for
adjacent-hypotenuse relationships Two wrinkles
occur When finding heights on top of heights
(ex. 6, pg 526) When finding heights with two
sightings (ex. 7, pg 526)
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Right Triangles Type 1One side, One angle
The Two Sightings Problem (Like example 7, pg
526)
Problem Were given two angles of two different
triangles and a difference in their common
side. Find The height (opposite side)
Method Solve two simultaneous equations for
tangent
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Right Triangles Type 1One Side, One Angle
The Two Sightings Problem Continued
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Right Triangles Type IITwo Sides
When given two sides, we use the Pythagorean to
find the third side, then use an appropriate
inverse trig function to find an angle.
A variation on this is the Bearing of an Airplane
Problem. This requires us to understand that
bearing is an angle measured CW from due North
(90 degrees).
A bearing of 55 degrees (NOT 35 degrees!)
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Right Triangles Type IITwo Sides
The Bearing of an Airplane Problem (Like Example
8, pg 527)
Problem Were given a bearing, a side, a
rotation and another side. Find The new
bearing after the rotation and after traveling
along the second side.
Method Find an angle wrt North, then find
another angle using tangent.
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Right Triangles Type IITwo Sides
Bearing of an Airplane Problem continued
A plane has a bearing of 55 degrees for 5 miles
and then turns NW 90 degrees for 14 miles. What
is the new bearing and what bearing should be
used to locate the plane?
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Right Triangles Type IITwo Sides
Bearing of an Airplane Problem Continued
The new bearing is the difference of old and 90
degrees, subtracted from 360.
To locate, we must work from the original axes.
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Right Triangles Type IITwo Sides
Bearing of an Airplane Problem continued
To locate the plane, we use the tangent and solve
for the bottom left angle.
This angle includes the 55 degrees, so we
subtract it out to get a bearing.
Because its CCW, we call it negative, or
subtract from 360 degrees to get 344.7 degrees.
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Non-Right Triangles
To solve non-right (oblique) triangles we must
know three pieces of information. This can
happen in five ways ASA, SAA, SSA, SAS, SSS
Note AAA is not sufficient The first three
cases can be solved using the law of sines. The
last two require the law of cosines. We will look
at the law of sines and then solve an SSA problem
because it requires us to evaluate three
potential solutions. The others are
straightforward.
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Law of Sines
The ratio of the sine of an angle to opposite
side is the same for each angle in a triangle.
We will do the proof on the board. A proof is
also available online at http//www.math.lsu.edu
/courses/1022/lecture/files/trig_8_2bw.pdf If you
dont like this site, google law of sines
proof and pick one of the 18,500 hits you do
like. The LSU site also gives good example
problems, which is why I include it here.
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Law of SinesSSA
The problem with SSA is the same problem
encountered when trying to show triangle
congruence. Namely, SSA can result in three
different answers no solutions, one solution or
two solutions. This depends on the length of the
second side relative to the first side.
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Law of SinesSSA
Too short In this case the law of sines will
give us a No Solution the value of the sine will
have to be bigger than 1 Just right In this
case, the law of sines will give us two angles,
one of which will make the sum of the angles
bigger than 180 degrees. The other angle is then
the correct one. Two big In this case, the law
of sines will give us two angles, both of which
keep the sum of the angles less than 180 degrees.
Both answers are ok and yield two triangles.
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Law of SinesSSA Two Big
Solve the triangle.
Given
Solution
Because the sine is positive in Q1 and Q2, two
potential solutions result.
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Law of SinesSSA Two Big
Both of these can add to 35 (given angle) and
give an answer less than 180. Therefore, both
are correct.
Triangle 1
Triangle 2
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Law of SinesSummary

• The ratios of opposite sides to sines of angles
  is the same for a given triangle
• Use Law of Sines when given ASA, AAS or SSA
• When solving SSA, dont try to draw the triangle,
  let the Law of Sines find the solution for you
• When solving for angles, use inverse sine

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Law of Cosines
The law of cosines is basically the pythagorean
theorem with a correction factor, to account for
a non-right angle. Because this correction
applies regardless of what angle we call
non-right, the Law of Cosines can be used to
find any side as is, which makes it more robust
than the pythagorean (which must be manipulated
to solve for any side other than the
hypotenuse). Key point The side we are solving
for is opposite the angle used in the formula.
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Law of Cosines
Proof http//mcraefamily.com/MathHelp/GeometryCo
sines.htm
This proof is nice because it shows you how to
quickly derive the law from the pythagorean
theorem, without a lot of foo-foo.
Notes 1. If the angle is obtuse, the correction
becomes positive, thus giving a longer c than
in a right triangle. 2. If the angle is right,
the correction disappears. 3. This can be used
as is to solve for a or b as well.
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Law of CosinesExample Problem
Note if you get a number bigger than 1, when
solving for cosine, you did something wrong or
the triangle is impossible. Note Do not
rearrange to solve for other angles! Use as is!
Given
Find The angles
Solution
The other angles solve similarly
Check do the angles add to 180?
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Law of CosinesSummary

• Use with SSS or SAS
• Also works with SSA, but much tougher than Law of
  Sines
• Decide which case you have, for a particular
  story problem
• Use as is, no rewriting necessary

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Areas of Triangles

• Area is easily found when we know a height
  (perpendicular to base).
• If height is not given, we can use the sine
  function to find it (to find any height, use sine
  of an angle opposite height)SAS
• If three sides are given, we use Herons
  FormulaSSS

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Area of an SAS Triangle
The area is
The height can be found using sine
Substitution gives
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Area of an SAS Triangle
Note if we are given side c, we simply use sine
of alpha and c. Similarly, we can drop a height
from either of the other two angles and use the
appropriate sine pg. 550 lists all formulae.
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Area of a SSS Triangle
In this case, we could use the law of cosines to
find an angle, then use the standard formula for
area. However, this is a lot of work that is
algorithmic in nature. That is, its all the
same process, just with different numbers. Is
there a way to run the process, arrive at some
answer and then plug in numbers? Yes. Its
called Herons Formula. Well do the proof on
the board. Its also available at
http//jwilson.coe.uga.edu/emt725/Heron/Trig.Heron
.html although this is a little thick, its
slick.
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Area of an SSS Triangle
The key to using Herons Formula is to remember
what s is. If you do this, solving SSS areas is
a 30 second affair.
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Sine Curves and Fitting

• Remember, a cosine is a shifted sine, so were
  actually talking about the same thing.
• Vocab
• Amplitude (A) height multiplies the sine
• Period (T) interval of repetition multiplies
  the argument
• Phase Shift where are the zeros? adds to the
  argument
• Rules for shifts, etc. still apply!

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Sine Curves
A is the amplitude. Omega is a frequency
constant. Period and frequency are related as
inverses on the interval 2pi, so
How can we shift the graph? By adding something
to the argument of the function.
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Sine Curves
The period will begin when the argument equals
zero. This means the frequency and shift work
together to change the period and where it
starts. The period starts at
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Sine Curves
Phi is coupled to omega, which means it isnt a
pure shift. The expression for x, not phi, is
the pure shift of the graph. That is, when x
is phi/omega, the function has zero as its value.
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Sine Curves
Amplitude Period Phase Shift
Examples p. 568 6,54,72 nothing by hand!
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Harmonic Motion

• No, not the marching band
• Motion that comes back on itself, repeats, has
  a frequency or period
• Examples
• Swing
• Spring
• Pendulum
• Described by a sine curve

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Harmonic Motion - Simple
Basically, this never happens. Simple Harmonic
Motion is periodic forever and ever. Think A
pendulum that goes forever. Impossible. Buta
good intro, so
Notes 1. Use cos when initial displacement is
not zero (pulling a spring, starting a
pendulum). Otherwise, you must use a phase
shift with sine why?. 2. a is the
amplitude, omega is the frequency constant
and t is time, usually in seconds.
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Harmonic Motion - Simple
Example A pendulum is started by lifting the bob
10 cm vertically from the rest position. The
pendulum oscillates with a period of .24 seconds.
What equation models this pendulum?
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Harmonic Motion - Simple

• Notes
• The book leaves everything in terms of pi.
  Although this is good for getting a reaction out
  of Dr. Kahler, we will use numerical (decimal)
  answers.
• Examples 1 and 2 on pg 577 break it down nicely.
• Remember this never really happens!

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Harmonic Motion - Damped
This is what really happens
Notice that the amplitude changes (decreases)
over time. We say the function, or motion,
decays or is damped. Reason usually friction.
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Harmonic Motion - Damped
This function decays in an envelope, which is an
exponential. If we graph the exponential on top
of this function, we see it is tangent at the
local maxima and minima.
This function must include an exponential and
a cosine.
with a continuously changing amplitude
and constant but modified period.
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Harmonic Motion Damped
Friction is the big meany here. The amount of
friction in an oscillating system depends on the
mass of the system, the atomic interactions at
touch points, etc. With the exception of mass,
all of this can be rolled into a constant called
the damping coefficient. Mass is dealt with
separately. As the value of the damping
coefficient goes up, the system should decay
faster. However, a larger mass will provide more
inertia to the system and tend to keep it in
motion, despite the damping. Therefore, mass
should impact the motion in an inverse way.
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Harmonic Motion Damped
Initial Amplitude a Decrease over
time Frequency constant
(modified) Constant over time Mass m Large
small decay Damping Coefficient d Large
large decay Envelope e Guts cosine
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Harmonic Motion Damped
Envelope

• A large b produces a large decay.
• A large m mitigates the decay.
• A maximum occurs when the (junk) is 0 or 2pik.
• A minimum occurs when the (junk) is an odd
  multiple of pi.

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Harmonic Motion Damped
Guts

• The cosine varies with time, t. The radical
  gives a constant.
• When the time multiplies the radical to give a
  multiple of pi, we get a max or min.
• Without the (junk) this would give a pure
  (simple) harmonic motion with a period of

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Harmonic Motion Damped
Putting it all together

• What happens to the envelope as t goes to
  infinity?
• What happens to the motion as m goes to infinity?
• What happens to the motion as m goes to zero?
• What happens to the motion as b goes to infinity?
• What happens to the motion as b goes to zero?

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Harmonic Motion Damped

• As t goes to infinity, the envelope goes to zero,
  thus the displacement goes to zero it stops.
• As m goes to infinity, the envelope goes to one
  and the damping disappears. The period goes to
  2pi/omega and we have SHM an infinite mass goes
  forever.
• As m goes to zero, the envelope goes to infinity
  and the period goes to zero. Thus, the motion is
  immediately damped zero mass systems will not
  oscillate
• As b goes to infinity, the envelope goes to
  infinity and the period goes to zero. Thus, the
  motion is immediately damped infinitely damped
  systems will not oscillate.
• As b goes to zero, the envelope goes to one and
  the damping disappears. The period goes to
  2pi/omega and we have SHM undamped systems
  will go forever.

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Harmonic Motion

• Simple or Damped
• Sine or Cosine, depending on where the
  displacement starts
• Damped is real life
• Check out crwash and go to Ch.7 Sinusoidal in
  the resources list for a summary.