Advanced Algebra Chapter 13

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Advanced Algebra Chapter 13

• Trigonometric Ratios and FUNctions

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Right Triangle Trigonometry13.1
3
Getting Started

• The Greek Alphabet!

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Getting Started

• Right Triangles
• Sides
• Hypotenuse
• Adjacent
• Opposite
• Angles
• Right Angle
• Theta
• Other angle
• The sum of all angles in any triangle is

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Getting Started

• What else do we know about right triangles?
• There was a guy with a theorem!

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Trig

• Six Trig functions
• Sine
• Cosine
• Tangent
• Cotangent
• Cosecant
• Secant

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Trig FUNctions
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Soh Cah Toa
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Trig FUNctions
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Evaluate all six trigonometric FUNctions for the
given triangle
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Evaluate all six trigonometric FUNctions for the
given triangle
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Evaluate all six trigonometric functions for the
given triangle
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Angles of Elevation or Depression

• Elevation
• Looking Up
• Depression
• Looking Down

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Angles

• A support cable from a radio tower makes an angle
  of 56 degrees with the ground. If the cable is
  250 feet long, how far above the ground does it
  meet the tower?

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Angles

• An airplane flying at 20,000 feet is headed
  toward an airport. The landing systems sends
  radar signals from the runway to the airplane,
  recording a 5 degree angle of elevation. About
  how many miles is the airplane from the runway?

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General Angles and Radian Measure13.2
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Define

• Angle
• Formed by two rays sharing a common endpoint
  known as the vertex

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Standard Position

• Standard Position
• Initial Side
• Terminal Side
• Positive v. Negative

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Angles

• 210 degrees

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Angles

• -45 degrees

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Angles

• 30 degrees
• -330 degrees
• 390 degrees
• Coterminal Angles

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Finding Coterminal Angles

• Two angles are coterminal iff one angle can be
  found my adding or subtracting multiples of 360
  degrees

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Coterminal Angles

• Find 2 positive and 2 negative angles coterminal
  to the following
• 70 degrees
• 115 degrees
• -5 degrees

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Circles

• What if we think about distance around a circle
  as a total of its angles?
• Circumference of a circle
• So,

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Circles
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The Unit Circle
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Converting from one to another

• 1 Radian is how many degrees
• Rewriting Degrees as Radians
• Rewriting Radians as Degrees

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Converting

• Convert 110 degrees to radians

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Converting

• Convert radians to degrees

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Why Radians?

• Work great with circles
• Already in terms of circumference so finding arc
  length is easy
• Whats arc length?
• Arc length (s) is the distance around a portion
  of a circle called a sector
• Whats a sector?
• Is a region of a circle bounded by two radii and
  the arc of the circle
• The angle formed is called the central angle

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Arc Length and Area of a Sector

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Trig FUNctions of Any Angle13.3
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Consider our Unit Circle again

• Make a list of all of the x- and y-values
• What do you notice about these?
• What about pos/neg?
• Well come back to this

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Consider our Unit Circle again

• Could we find a short cut to this stuff?
• In other words, do we need the entire circle?

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Consider our Unit Circle again

• Everything we need happens in the 1st quadrant!
• So, we use reference angles
• Reference angle
• Acute angle formed by the terminal side of the
  angle and the x-axis
• i.e. use the shortest distance to the x-axis as
  your reference angle

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Find the reference angles for

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Why reference angles and ordered pairs?

• When on the unit circle
• Y-values Sine values!
• X-values Cosine values!

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Define trig functions as





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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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Using reference angles

• Find the sin of the following

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Using reference angles

• Find cos of the following?

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Using reference angles,

• Find tan of the following

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Inverse Trig FUNctions13.4
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Domain and Range

• Domain
• The numbers you plug IN to a function
• Range
• The numbers you get OUT of a function

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Inverse FUNctions

• To get an inverse, we switch the domains and
  ranges!
• For our new function
• The old range becomes our new domain
• The old domain becomes our new range

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Inverse FUNctions

• Consider the sine function
• Domain
• Range
• How about inverse sine or
• Domain
• Range

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Inverse FUNctions

• Consider the cosine function
• Domain
• Range
• How about inverse cosine or
• Domain
• Range

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Inverse FUNctions

• Consider the tangent function
• Domain
• Range
• How about inverse tangent or
• Domain
• Range

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Inverse

• Evaluate the following expression

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Inverse

• Evaluate the following expression

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Inverse

• Evaluate the following expression

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Inverse

• Solve the equation where

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Inverse

• Solve the equation where

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The Law Of Sines13.5
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Why?

• Allows us to find missing parts of triangles that
  are NOT right triangles
• Often more practical then breaking triangles down
  into separate parts to get right angles

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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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The Law of Sines
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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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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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Area of a Triangle

• The area of any triangle is given by one half the
  product of the lengths of any two sides times the
  sine of their included angle

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Area of a triangle
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Area of a triangle
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The Law of Cosines13.6
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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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The Law of Cosines
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Law of Sines or Cosines?

• Sines is much easier, however we cant use it all
  the time
• Use law of cos when you have to
• Otherwise, you can use law of sines

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Herons Formula

• The area of the triangle with sides of length a,
  b, and c iswhere
• The variable s is called the semiperimeter, or
  the half perimeter of the triangle

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Herons Formula
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