College_Trigonometry_Ch7

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					           Chapter 7
Applications of Trig and Vectors
Section 7.1 Oblique Triangles & Law of Sines
Section 7.2 Ambiguous Case & Law of Sines
Section 7.3 The Law of Cosines
Section 7.4 Vectors and the Dot Product
Section 7.5 Applications of Vectors
  Section 7.1 Oblique Triangles &
           Law of Sines
• Congruency and Oblique Triangles
• Law of Sines
• Solving using AAS or ASA Triangles
Congruency and Oblique Triangles
• If we use A for angles and S for sides what
  are all of the three letter combinations you
  could create?

• Which of these can we use to prove the
  triangles are congruent?
      Congruence Shortcuts


ASA
                             YES




SAA                          YES




                             NO


AAA
      Congruence Shortcuts


SSS
                             YES




SAS                          YES




                             NO


SSA
      Data required for Solving
         Oblique Triangles
1. One side and two angles (ASA or AAS)
2. Two sides and one angle not included
   between the two sides (ASS). Yep this
   one can create more than one triangle
3. Two sides and the angle between them
   (SAS)
4. Three sides (SSS)
5. Three angles (AAA) Yep this one only
   creates similar triangles.
                                      E
       Given:
          ~
       AR = ER
                         R
          ~
       EC = AC                                 C

       Show /_E ~ /_A
                =

                                          A

                ~
  1.        AR = ER

Given

                ~            ~                   ~
   2.       EC = AC     ΔRCE = ΔRCA           /_E = /_A

Given                     SSS                 CPCTC
                ~
   3.       RC = RC
Reflexive
                                        E
      Given:
      /_E ~ /_A
          =
           ~               R
      /_ECR = /_ACR                              C

      Show AR ~ ER
               =

                                            A

                 ~
          1. /_E = /_A

Given

                ~              ~                   ~
2.        / ECR = / ACR   ΔRCE = ΔRCA           AR = ER

Given                          AAS               CPCTC
                ~
     3.     RC = RC
Reflexive
                                        E
      Given:
      /_E ~ /_A
          =
           ~               R
      /_ERC = /_ARC                              C

      Show AR ~ ER
               =

                                            A

                 ~
          1. /_E = /_A


Given

                ~              ~                   ~
2.        / ERC = / ARC   ΔRCE = ΔRCA           AR = ER

Given                          ASA               CPCTC
                ~
     3.     RC = RC
Reflexive
              Law of Sines
In any triangle ABC, with sides a, b, and c,

     a       b       a       c       b       c
   sin A
         = sin B , sin A = sin C , sin B = sin C

This can be written in compact form as
              a       b       c
                  =       =
            sin A   sin B   sin C
          Area of a Triangle
In any triangle ABC, the area A is given by
  any of the following formulas:

               A = ½bc sin A
               A = ½ab sin C
               A = ½ac sin B
  Section 7.2 Ambiguous Case &
           Law of Sines

• Description of the Ambiguous Case
• Solving SSA Triangles (Case 2)
• Analyzing Data for Possible Number
         Ambiguous Case Acute
   Number of                  Condition Necessary for Case to Hold
                     Sketch
Possible Triangles
        0                                     a<h




        1                                     a=h



        1                                     a>b




        2                                    b>a>h
            Ambiguous Case
Number of               Condition Necessary
 Possible      Sketch     for Case to Hold
Triangles
    0                          a<b




    1                          a>b
              SSA Cases
• Remember since SSA results in two
  possible triangles we must check the
  angles supplement as well. So if we find
  the angle is 73 then we also have to
  check 180 – 73 = 107.
Section 7.3 The Law of Cosines
•   Derivation of the Law of Cosines
•   Solving SAS Triangles Case 3
•   Solving SSS Triangles Case 4
•   Heron’s Formula for the Area of a Triangle
Triangle Side Length Restriction
• In any triangle, the sum of the lengths of
  any two sides must be greater than the
  length of the remaining side.
Law of Cosines
            Law of Cosines
In any triangle ABC, with sides a, b, and c,


  a2  = + – 2bc cos A
          b2      c2

  b 2 = a2 + c2 – 2ac cos B

  c 2 = a2 + b2 – 2ab cos C
     Oblique Triangle Case 1
• One side and two angles AAS or ASA
  1. Find the remaining angle using the angle
     sum formula (A+B+C)=180
  2. Find the remaining sides using the Law of
     Sines
     Oblique Triangle Case 2
• Two sides and a non-included angle SSA
  1. Find an angle using the Law of Sines
  2. Find the remaining angle using the Angle
     Sum Formula
  3. Find the remaining side using the Law of
     Sines

  There may be no triangle or two triangles
     Oblique Triangle Case 3
• Two sides and an included angle SAS
  1. Find the third side using the Law of Cosines
  2. Find the smaller of the two remaining angles
     using the Law of Sines
  3. Find the remaining angle using the angle
     sum formula
     Oblique Triangle Case 4
• Three sides SSS
  1. Find the largest angle using the Law of
     Cosines
  2. Find either remaining angle using the Law of
     Sines
  3. Find the remaining angle using the angle
     sum formula
       Heron’s Area Formula
If a triangle has sides of lengths a, b, and c,
   and if the semi-perimeter is
                  s= ½(a+b+c)

  then the area of the triangle is
               A =  s(s-a)(s-b)(s-c)
         Section 7.4 Vectors and
            the Dot Product
•   Basic Vector Terminology
•   Finding Components and Magnitudes
•   Algebraic Interpretation of Vectors
•   Operations with Vectors
•   Dot Product and the Angle between
    Vectors
         Basic Terminology
• scalars – quantities involving only
  magnitudes
• vector quantities – quantities having both
  magnitude and direction
• vector – a directed line segment
• magnitude – length of a vector
• initial point – vector starting point
• terminal point – second point through which
  the vector passes
           Sum of vectors
To find the sum of two vectors A and B:
                     A+B
               resultant vector

                    or
        Difference of vectors
To find the difference of 2 vectors A and B:
                    A+(-B)
                resultant vector

                     or
              Scalar Product
To find the product of a real number k and a
 vector A : kA=A+A+…+A (k times)

Example: 3A
       Magnitude and Direction
       Angle of a Vector <a,b>
The magnitude of vector u=<a,b> is
given by
             |u| = a2 + b2

The direction angle  satisfies
                tan  =b/a,
where a ≠ 0.
       Horizontal and Vertical
            Components
The horizontal and vertical components,
 respectively, of a vector u having
 magnitude |u| and direction angle  are
 given by
                 = |u| cos 
                     and
                  =|u| sin 
          Vector Operations
For any real numbers a, b, c, d, and k,

        <a, b> + <c, d> = <a+c, b+d>
             k ·<a, b> = <ka, kb>
     If a = <a1, a2>, then -a = <-a1, -a2>
      <a, b> - <c, d> = <a, b> + -<c, d>
        Unit Vectors
     i = <1, 0>       j = <0, 1>




 i, j Form for Vectors
If v = <a, b>, then     v = ai + bj
           Dot Product
  The dot product of the two vectors
u = <a, b> and v = <c, d> is denoted by
    u·v, read “u dot v,” and given by

            u·v = ac + bd
  Properties of the Dot Product
For all vectors u, v, w and real numbers k


     u·v=v·u           u ·(v+w) = u·v + u·w

(u+v) ·w= u·w + v·w (ku)·v=k(u·v)=u·(kv)

      0·u=0                 u · u = |u|2
     Geometric Interpretation of
          Dot Product
If  is the angle between the two nonzero
   vectors u and v, where 0<  <180, then

             u·v = |u||v| cos 
        Orthogonal Vectors

Two nonzero u and v vectors are orthogonal
 vectors if and only if u · v = 0
Section 7.5 Applications of Vectors
• The Equilibrant
• Incline Applications
• Navigation Applications
               Equilibrant
A vector that counterbalances the resultant
  is called the equilibrant. If u is a vector
  then –u is the equilibrant.

                  u + -u = 0
                 Equilibrant Force
• Use the law of Cosines
                                                  B
                                                 130à      60
                                          48
                                                     v                 C
                         -v
                                      A
                                                                  48
                                            60

                                               The required angle can be
  |v|2 = 482+    – 2(48)(60)cos(130à)
               602                             found by subtracting angle CAB
  |v|2 ≈ 9606.5                                from 180à.
  |v| ≈ 98 newtons                                          98         60
                                                         sin 130à = sin CAB
                              CAB ≈ 28à so £ = 180à - 28à =152à
Inclined Application



    20à
          50
                C
          A x

sin 20à = |AC|/50
|AC| ≈ 17 pounds of force

				
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