Sine and Cosine Laws!

  The Review!
A



    c       b




B       a       C
A



    c          b
                          Sine Law

B         a        C




        SinA       SinB     SinC
      ...
A

                         A = 30°       B=?
     c         b         b=8           c=?
                         a=9     ...
A
                    Cosine Law
    c       b




B       a       C




         2   2   2
        a = b + c - 2bc cosA
A

                         A = 30°   B=?
      c         b
                         b=8       a=?
                       ...
Draw triangle ABC such that
∠A = 45°
a=6
b=7
Draw triangle ABC such that
∠A = 45°                       B
a=6                        c       a
b=7
                    ...
But.....

...is there another possible triangle?
Draw triangle ABC such that
∠A = 45°
a=6
b=7



                 B       a
             c
         A               C
     ...
B
            a

    a
A               C
B
                a
    B
        a
A                   C
B1
    B2            a

         a
A                     C
B1 a
    B1
    a
A               C
B1 a
       B1
     B2 a
A                   C

    B2 = 180 - B1
B2
        c           a
    A                   C
             b
     B2 = 180 - B1
     B2 = 180 - 55.6 = 124.4

  C2 = ...
This is called the Ambiguous Case. This
means that the information isn't perfectly
clear. The information leads to more th...
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Ambiguous Case1

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Ambiguous Case1

  1. 1. Sine and Cosine Laws! The Review!
  2. 2. A c b B a C
  3. 3. A c b Sine Law B a C SinA SinB SinC = = a b c
  4. 4. A A = 30° B=? c b b=8 c=? a=9 C=? B a C sin30 = sinB B = 26.4° 9 8 C = 180 - 30 - 26.4 = 125.4° c = 9 c = 14.6 sin125.9 sin30
  5. 5. A Cosine Law c b B a C 2 2 2 a = b + c - 2bc cosA
  6. 6. A A = 30° B=? c b b=8 a=? c=9 C=? B a C 2 2 2 a = 8 + 9 - 2(8)(9) cos 30 a = 4.5 You can then use the sine law to find the rest
  7. 7. Draw triangle ABC such that ∠A = 45° a=6 b=7
  8. 8. Draw triangle ABC such that ∠A = 45° B a=6 c a b=7 A C b sin45 = sinB B = 55.6 6 7 C = 180 - 45 - 55.6 = 79.4 c = 6 c = 8.34 sin79.4 sin45
  9. 9. But..... ...is there another possible triangle?
  10. 10. Draw triangle ABC such that ∠A = 45° a=6 b=7 B a c A C b
  11. 11. B a a A C
  12. 12. B a B a A C
  13. 13. B1 B2 a a A C
  14. 14. B1 a B1 a A C
  15. 15. B1 a B1 B2 a A C B2 = 180 - B1
  16. 16. B2 c a A C b B2 = 180 - B1 B2 = 180 - 55.6 = 124.4 C2 = 180 - 124.4 - 45 = 10.6 c = 6 c = 1.56 sin10.6 sin45
  17. 17. This is called the Ambiguous Case. This means that the information isn't perfectly clear. The information leads to more than 1 possible solution. Which is correct? Since there isn't any more information given in the question to help us determine which is the correct triangle we say both are correct, since both are possible.

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