JEE Main & Advanced Mathematics Trigonometric Equations Area of Triangle

Area of Triangle

Category : JEE Main & Advanced

Let three angles of \[\Delta ABC\] are denoted by \[A,\,\,B,\,\,C\] and the sides opposite to these angles by letters \[a,\,\,b,\,\,c\] respectively.

 

 

(1) When two sides and the included angle be given :

 

 

The area of triangle ABC is given by,

 

 

\[\Delta =\frac{1}{2}bc\sin A=\frac{1}{2}ca\sin B=\frac{1}{2}ab\sin C\]

 

 

i.e., \[\Delta =\frac{1}{2}\] (Product of two sides) \[\times \] sine of included angle

 

 

(2) When three sides are given :

 

 

Area of \[\Delta ABC=\,\Delta =\sqrt{s(s-a)(s-b)(s-c)}\]

 

       

where semiperimeter of triangle \[s=\frac{a+b+c}{2}\]

 

 

(3) When three sides and the circum-radius be given :

 

 

Area of triangle\[\Delta =\frac{abc}{4R}\], where R be the circum-radius of the triangle.

 

 

(4) When two angles and included side be given :

 

 

\[\Delta =\frac{1}{2}{{a}^{2}}\frac{\sin B\sin C}{\sin (B+C)}=\frac{1}{2}{{b}^{2}}\frac{\sin A\sin C}{\sin (A+C)}=\frac{1}{2}{{c}^{2}}\frac{\sin A\sin B}{\sin (A+B)}\]

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