Pre Calc Chapters 5 and 6

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Pre Calc Chapters 5 and 6

• Trigonometric Functions of Real Numbers

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The Unit Circle5.1
Q What do you get if you cross a mountain
climber with a mosquito?
A Nothing, you cant cross a scalar with a
vector
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The Unit Circle

• The Unit Circle of radius 1 centered at the
  origin in the xy-plane. Its equation is

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The Unit Circle
Standard Position
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The Unit Circle

• Is the point is on the unit
  circle??

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The Unit Circle
Measured in Radians
Why does 2p360
Think Circumference!
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The Unit Circle
Moving around the unit circle
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The Unit Circle
Terminal Point The point P(x , y) obtained by
traveling from the point (1 , 0)
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Reference Points

• The reference number is the shortest distance
  along the unit circle between the terminal point
  and the x-axis
• Makes finding the corresponding ordered pairs
  easier to find!

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Reference Points
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Reference Points
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Reference Points
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p.416 1-3, 5-7, 11-18, 23-25
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Trigonometric Functions of Real Numbers5.2
Three statisticians went duck hunting. A duck was
approaching and the first statistician shot, and
missed the duck by being a foot too high. The
second shot and was a foot too low. The third
cried, "We hit it!"
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Trig Functions
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Trig Functions
Soh Cah Toa
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Trig Functions

• Let
• Find sin(t)
• Find cos(t)
• Find tan(t)

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Trig Functions
Let t be any real number and let P(x,y) be a
point on the unit circle We can then define our
trig functions as follows
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Trig Functions
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Trig Functions


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Trig Functions Using the Unit Circle
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Trig Functions Using the Unit Circle
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Trig Functions Using the Unit Circle
Using the point
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Domain of Trig Functions

• Sine, cosine
• All Real Numbers
• Tangent, Secant
• All Real Numbers except for any
  integer n
• Cotangent, Cosecant
• All real numbers other than for any
  integer n

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Signs of Trig Functions
Quadrant Positive Functions Negative Functions
I All None
II Sin, csc Cos, sec, tan, cot
III Tan, cot Sin, csc, cos, sec
IV Cos, sec Sin, csc, tan, cot
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Signs of Trig Functions
All Students Take Calculus
All
Sine
Tells us which values are positive
Tangent
Cosine
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Even and Odd Functions

• Even
• Sine, Cosecant, Tangent, and Cotangent
• Odd
• Cosine and secant

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SoWhat does that mean?

• Tells us the sign of each function, based on the
  quadrant it is in!
• Ex consider
• Sin, cos, tan, etc.

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p.426 3-22, 27-29
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Trig Functions5.3

• There are 10 types of people in this world
• those who understand binary and those who
  dont

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Graphing Trig Functions
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Periodic Functions

• A function f is periodic iff there is a positive
  number p such that for
  every t
• The smallest of these numbers p is called the
  period of the function

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Periodic Functions

• Think terminal points
• every 2pi units around the circle, you are at
  the same point so the function evaluated at those
  points will be the same

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Sine and Cosine Curves

• These two functions are often referred to as
  sinusoidal curves

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Sine and Cosine Curves

• General form of sine and cosine
• Amplitude
• Period
• phase shift b
• Vertical shift c

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Sine Functions

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Cosine Functions

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Can sin(x) be made to look like cos(x)?
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p.439 1-25 Odd
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Trig Functions and Asymptotes5.4

• Q How do we know the fractions are
  all European?
• A Because they are all over Cs!

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Tangent and Cotangent Graphs

• Tangent and Cotangent both have a period of
• In other words

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Where Do Asymptotes Occur?

• At what values can Tan not be evaluated?
• So we can say

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Tangent
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Cot(x)
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Modifying Graphs of Tan and Cot

• Period
• Amplitude
• Phase Shift b
• Vertical Shift c

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Cosecant and Secant

• csc and sec graphs have a period of
• In other words

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Csc(x)
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Sec(x)
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Chapter 6 Angle Measure
Q What does trigonometry have in common with a
beach?
A Tan Gents
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Angle Measure
Standard Position
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Angle Measure
Measure
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Angle Measure Corresponding Angles
Coterminal
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Angle Measure Radians to Degrees
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Arc Length
Angle MUST be in radians!!!!!
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Arc Length
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Area of a Sector
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p.4801-4, 9-12, 17-19, 23-26, 29-31, 41-49, 51-58
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Q) If a person sits on a cot, why will he get a
sunburn?A) Because, if a person is sitting on
cot, he will be one upon cot and hence is tan.
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Trigonometry of Right Triangles
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Trigonometry of Right Triangles
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Applications
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Applications
Find the height of large objects -Trees -Buildin
gs -Watertowers -etc.
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Applications

• Angle of elevation
• Angle from level up to line of sight
• Angle of Depression
• Angle from level down to line of sight

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Applications
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Applications
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• p.4891-21 Odd, 38-40

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(more) Definitions of Trig Functions
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All Students Take Calculus
Reference Angle
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Finding Exact Values
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p.5011-4, 7-18, 31-34
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Fundamental Identities (Part 1)
Reciprocal Identities
Pythagorean Identities
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Fundamental Identities

• Why???
• Allows us to manipulate problems
• Eventually makes them easier to work with
• Allows us to see similarity of problems
• i.e. get everything in terms of sine

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Modifying Fundamental Identities

• We can modify any of the identities to get what
  we want

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Using Fundamental Identities

• Write sine in terms of cosine in Q III
• Write cosine in terms of tangent in Q I

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Area of a Triangle
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Area of a Triangle

• Find the area of the circle outside of the
  triangle

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p.50135-45, 51-54
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Solving Oblique Triangles
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Why SSA??

• Cant use for congruence or similar triangles BUT
  we can use SSA using the law of sines
• SoWe need to be careful!

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No SSSNot yet anyway

• With Law of Sines we need to know at least one
  angle
• The Law of CosinesComing soon ?

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Law of Sines

• In any triangle the lengths of the sides are
  proportional to the sines of the corresponding
  opposite angles

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Law of Sines
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Law of SinesThe Ambiguous Case
2 solutions
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Law of SinesThe Ambiguous Case

• No Solution

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Law of SinesThe Ambiguous Case
Unique Solution
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The Ambiguous Case

• When can it exist?
• When the given angle is less than 90 degrees
• Ex Angle A 40 degrees, side a 18, side b 25

So B 63 or B 180-63 117
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The Ambiguous CaseCase 1
Angle B is 63 degrees so
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The Ambiguous CaseCase 2
Angle B is 117 Degrees
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p.5101-17 Odd
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The Law of Cosines
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The Law of Cosines

• Advantage of The Law of Cosines?
• Do not need to know an angle
• NOW we can use SSS

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The Law of Cosines
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The Law of Cosines
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Herons Formula

• S is the semi perimeter
• Half the perimeter of the triangle
• Advantage??
• Do not need an angle measure
• Very practical

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Herons Formula
Find the area of the following triangle?
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p.5181-15 Odd, 27-31 All