Basic trigonometry question

  1. We got a circle with a radius R.

    From a distance D from the centerpoint a line is inserted at an offset angle A1 from a line drawn though the centerpoint C of the circle, see the picture below.


    I would like the are of the red triangle, provided D, R and A1.

    I drew another triangle with one of its corner in the circles center for help, extracted the angle A2, got H and could then solve the problem but. However I wonder if there is a neater way than this.

    What I did:

    A3 = 180°-A1

    Law of cosines give

    sin (A3) / R = (sin A4) /D

    A4 = arcsin( sin(A3) * D/R ) = arcsin( sin(180°-A1) * D/R)

    A2 = 180° - A3 - A4 = 180° - (180°-A1) - arcsin( sin(180°-A1) * D/R) =

    = A1 - arcsin( sin(A1)* D/R)

    sin(A1) = H/R

    H= R*sin(A1)

    cos(A2) = (B+D)/R

    B= R*cos(A2) -D

    The red area = B*H/2 = (R*cos(A2)-D)*R*sin(A2)/2

    And so forth. However, I get the feeling that this solution is more complicated than necessary?

    EDIT: Btw the circle hasn't got much to do with the problem, but I'm just using it in a next step.
  2. jcsd
  3. SteamKing

    SteamKing 10,929
    Staff Emeritus
    Science Advisor
    Homework Helper

    What you call the Law of Cosines is actually the Law of Sines.
  4. Typo.

    More suggestions?
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