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.

The Law of Cosine

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

The Law of Sine

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

The Law of Tangent

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

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 $s=\dfrac{a+b+c}{2}$.

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 $s=\dfrac{a+b+c}{2}$.

Circumradius

Assume $R$ to be circumradius:

  • $R=\dfrac{a}{2\sin\alpha }=\dfrac{b}{2\sin\beta }=\dfrac{c}{2\sin \gamma }$
  • $R=\dfrac{abc}{4\Delta}$

Inradius

Assume $r$ to be inradius

  • $r=\dfrac{\Delta}{s}$

Escribed Circle

  • $r_1=\dfrac{\Delta }{s-a}$
  • $r_2=\dfrac{\Delta }{s-b}$
  • $r_3=\dfrac{\Delta }{s-c}$

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