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Basic trigonometry question

  1. Feb 23, 2014 #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. Feb 23, 2014 #2


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    Homework Helper

    What you call the Law of Cosines is actually the Law of Sines.
  4. Feb 26, 2014 #3

    More suggestions?
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