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
- a2=b2+c2−2bccosα
- b2=c2+a2−2cacosβ
- c2=a2+b2−2abcosγ
- cosα=b2+c2−a22bc
- cosβ=c2+a2−b22ac
- cosγ=a2+b2−c22ab
The Law of Sine
- asinα=bsinβ=csinγ
The Law of Tangent
- a−ba+b=tan(α−β2)tan(α+β2)
- b−cb+c=tan(β−γ2)tan(β+γ2)
- c−ac+a=tan(γ−α2)tan(γ+α2)
Half Angles Formulas
- sinα2=√(s−b)(s−c)bc
- sinβ2=√(s−c)(s−a)ca
- sinγ2=√(s−a)(s−b)ab
- cosα2=√s(s−a)bc
- cosβ2=√s(s−b)ca
- cosγ2=√s(s−c)ab
- tanα2=√(s−b)(s−c)s(s−a)
- tanβ2=√(s−c)(s−a)s(s−b)
- tanγ2=√(s−a)(s−b)s(s−c),
where s=a+b+c2.
Area of Triangle
Assume Δ to be area of triangle.
- Δ=12bcsinα=12casinβ=12absinγ
- Δ=a2sinβsinγ2sinα=b2sinγsinα2sinβ=c2sinαsinβ2sinγ
- Δ=√s(s−a)(s−b)(s−c), (Heron’s Formula),
where s=a+b+c2.
Circumradius
Assume R to be circumradius:
- R=a2sinα=b2sinβ=c2sinγ
- R=abc4Δ
Inradius
Assume r to be inradius
- r=Δs
Escribed Circle
- r1=Δs−a
- r2=Δs−b
- r3=Δs−c