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Solution and Area of Oblique Triangle

These are the common formulas used in Chapter 12 of Textbook of Algebra and Trigonometry Class XI, Punjab Textbook Board Lahore. This handout is very helpful to remember the formulas. All these formulas are given for real valued and defined trigonometric functions. A PDF file can be downloaded for high quality printing and a word file is also given if you wish to modify the contents or credit as you need. <panel title=“The Law of Cosine”>

  • $a^2=b^2+c^2-2bc\cos \alpha$
  • $b^2=c^2+a^2-2ca\cos \beta$
  • $c^2=a^2+b^2-2ab\cos \gamma$

  • $\cos\alpha =\dfrac{b^2+c^2-a^2}{2bc}$
  • $\cos\beta =\dfrac{c^2+a^2-b^2}{2ac}$
  • $\cos\gamma =\dfrac{a^2+b^2-c^2}{2ab}$

</panel> <panel title=“The Law of Sine”>

  • $\dfrac{a}{\sin \alpha }=\dfrac{b}{\sin \beta }=\dfrac{c}{\sin \gamma }$

</panel> <panel title=“The Law of Tangent”>

  • $\dfrac{a-b}{a+b}=\dfrac{\tan \left( \tfrac{\alpha -\beta }{2} \right)}{\tan \left( \tfrac{\alpha +\beta }{2} \right)}$

  • $\dfrac{b-c}{b+c}=\dfrac{\tan \left( \tfrac{\beta-\gamma}{2} \right)}{\tan \left( \tfrac{\beta+\gamma}{2} \right)}$

  • $\dfrac{c-a}{c+a}=\dfrac{\tan \left( \tfrac{\gamma -\alpha}{2} \right)}{\tan \left( \tfrac{\gamma +\alpha}{2} \right)}$

</panel> <panel title=“Half Angles Formulas”>

  • $\sin\dfrac{\alpha}{2}=\sqrt{\dfrac{\left(s-b \right)\left(s-c \right)}{bc}}$
  • $\sin\dfrac{\beta}{2}=\sqrt{\dfrac{\left(s-c \right)\left(s-a \right)}{ca}}$
  • $\sin\dfrac{\gamma}{2}=\sqrt{\dfrac{\left(s-a \right)\left(s-b \right)}{ab}}$

  • $\cos\dfrac{\alpha}{2}=\sqrt{\dfrac{s\left(s-a \right)}{bc}}$
  • $\cos\dfrac{\beta}{2}=\sqrt{\dfrac{s\left(s-b \right)}{ca}}$
  • $\cos\dfrac{\gamma}{2}=\sqrt{\dfrac{s\left(s-c \right)}{ab}}$

  • $\tan\dfrac{\alpha}{2}=\sqrt{\dfrac{(s-b)(s-c)}{s(s-a)}}$
  • $\tan\dfrac{\beta}{2}=\sqrt{\dfrac{(s-c)(s-a)}{s(s-b)}}$
  • $\tan\dfrac{\gamma}{2}=\sqrt{\dfrac{(s-a)(s-b)}{s(s-c)}}$,

where $s=\dfrac{a+b+c}{2}$. </panel> <panel title=“Area of Triangle”> Assume $\Delta$ to be area of triangle.

  • $\Delta=\dfrac{1}{2}bc\sin\alpha =\dfrac{1}{2}ca\sin \beta =\frac{1}{2}ab\sin \gamma $
  • $\Delta=\dfrac{a^2\sin\beta\sin\gamma }{2\sin\alpha }=\dfrac{b^2\sin\gamma\sin\alpha}{2\sin\beta}=\dfrac{c^2\sin\alpha\sin \beta}{2\sin\gamma }$
  • $\Delta=\sqrt{s( s-a)(s-b)(s-c)}$, (Heron’s Formula),

where $s=\dfrac{a+b+c}{2}$. </panel> <panel title=“Circumradius”> Assume $R$ to be circumradius:

  • $R=\dfrac{a}{2\sin\alpha }=\dfrac{b}{2\sin\beta }=\dfrac{c}{2\sin \gamma }$

  • $R=\dfrac{abc}{4\Delta}$

</panel> <panel title=“Inradius”> Assume $r$ to be inradius

  • $r=\dfrac{\Delta}{s}$

</panel> <panel title=“Escribed Circle”>

  • $r_1=\dfrac{\Delta }{s-a}$

  • $r_2=\dfrac{\Delta }{s-b}$

  • $r_3=\dfrac{\Delta }{s-c}$

</panel>

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