Trigonometric Properties of a Triangle – HSC Math

Posted In: Mathematics

  1. Sine Rule
    $a/{Sin\ A}=b/{Sin\ B}=c/{Sin\ C}$
  2. Cos Rule
    $cos\ A={b^2+c^2-a^2}/{2bc}$
    $cos\ B={a^2+c^2-b^2}/{2ac}$
    $cos\ C={a^2+b^2-c^2}/{2ab}$
    $a=b\ cos\ C+c\ cos\ B$
    $b=c\ cos\ A+a\ cos\ C$
    $c=a\ cos\ B+b\ cos\ A$
  3. Tan Rule
    $tan\ {B-C}/2={b-c}/{b+c}\ cot\ A/2$
    $tan\ {A-B}/2={a-b}/{a+b}\ cot\ C/2$
    $tan\ {C-A}/2={c-a}/{c+a}\ cot\ B/2$
    $tan\ A/2=√{(s-b)(s-c)}/√{s(s-a)}={(s-b)(s-c)}/{∆}$
    $tan\ B/2=√{(s-a)(s-c)}/√{s(s-c)}={(s-a)(s-c)}/{∆}$
    $tan\ C/2=√{(s-a)(s-b)}/√{s(s-c)}={(s-a)(s-b)}/{∆}$
    Here, $s={a+b+c}/2, ∆=$Area of that triangle.
  4. Area of a triangle
    $∆=1/2ah_a=1/2bh_b=1/2ch_c$
    $=1/2ab\ sin\ C=1/2bc\ sin\ A=1/2ca\ sin\ B$
    $={abc}/{4R}=sr$
    $=√{s(s-a)(s-b)(s-c)}$
    $=1/4√{2a^2b^2+2b^2c^2+2c^2a^2-a^4-b^4-c^4}$
    Where, R is the circum radius of the triangle.
    $r=4R\ sin\ A/2\ sin\ B/2\ sin\ C/2$
    Where, r is the in radius of the triangle.

Post Tags: HSC | Trigonometry


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