Law of Tangent

Discussion in 'Geometry and Trigonometry' started by nycmathguy, Feb 1, 2022.

  1. nycmathguy

    nycmathguy

    Joined:
    Jun 27, 2021
    Messages:
    5,386
    Likes Received:
    422
    In trigonometry, we are surely to come across the law of sines and law of cosines. However, there is a law of tangent that is never taught in trigonometry.

    1. Why is the law of tangents ignored?

    2. Solve the following using the law of tangents.

    In triangle ABC, little a = 321, little b = 234 and angle C = 71°. Find angle A, angle B and little c to the nearest tenth.

    Note: It is called the Law of Tangents not the Law of Tangent. I forgot the letter "s" in the title for this thread.
     
    nycmathguy, Feb 1, 2022
    #1
  2. nycmathguy

    MathLover1

    Joined:
    Jun 27, 2021
    Messages:
    2,989
    Likes Received:
    2,884
    upload_2022-1-31_20-45-38.png
    In triangle ABC, with sides a, b, and c opposite the respective angles A, B, and C, the law of tangents states:

    (a-b)/(a+b)=tan((A-B)/2)/tan((A+B)/2)….(1)

    Similarly for other sides,

    (b-c)/(b+c)=tan((B-C)/2)/tan((B+C)/2)….(2)

    (c-a)/(c+a)=tan((C-A)/2)/tan((C+A)/2) …..(3)


    if a = 321, b = 234 and angle C = 71°

    (234-c)/(234+c)=tan((B-71)/2)/tan((B+71)/2)
    c = 234 sin(71) (1/sin(B))...........(2a)

    (c-321)/(c+321)=tan((71-A)/2)/tan((71+A)/2)
    c = 321 sin(71) (1/sin(A))................(3a)

    from (2a) and (3a) we have

    234 sin(71) (1/sin(B))=321 sin(71) (1/sin(A))..........simplify

    234/sin(B)=321/sin(A)

    234sin(A)=321sin(B)

    sin(A)=(321/234)sin(B)
    sin(A)=(107/78)sin(B)
    A=sin^-1((107/78)sin(B))

    go to
    c = 321 sin(71) (1/sin(A))................(3a), substitute A
    c = 321 sin(71) (1/sin(sin^-1((107/78)sin(B))))
    c = 321sin(71)(107/78) sin(B)..........(3b)

    from (2a) and (3b) we have

    234 sin(71) (1/sin(B))=321sin(71)(107/78) sin(B).......solve for B

    234/sin(B)=321(107/78) sin(B)
    sin^2(B)=234/(321(107/78))
    sin^2(B)=6084/11449

    sin(B)=sqrt(6084/11449)

    sin(B)=0.7289719626168224

    B=sin^-1(0.7289719626168224)
    B=46.80027915261779°

    since given C = 71°, then angle A will be

    A=180-( 71°+46.80027915261779°)
    A=180°-( 117.80027915261779° )
    A=62.19972084738221°

    go to
    c = 321 sin(71) (1/sin(A)), substitute A
    c = 321 sin(71) (1/sin(62.19972084738221))
    c = 343.11414209938454

    angle A, angle B and little c to the nearest tenth.
    A=62.2°
    B=46.8°
    c = 343.1
     
    MathLover1, Feb 1, 2022
    #2
    nycmathguy likes this.
  3. nycmathguy

    nycmathguy

    Joined:
    Jun 27, 2021
    Messages:
    5,386
    Likes Received:
    422
    Wow! This is complicated. I can find angles A and B and side c without the law of tangents. So, I will stick to the law of sines and law of cosines. Why complicate my life? Right?
     
    nycmathguy, Feb 1, 2022
    #3
Ask a Question

Want to reply to this thread or ask your own question?

You'll need to choose a username for the site, which only take a couple of moments (here). After that, you can post your question and our members will help you out.
Similar Threads
Loading...