JEE Main & Advanced Mathematics Trigonometric Equations Some Important Results

(1)

\[a=h\,(\cot \alpha -\cot \beta )=\frac{h\sin (\beta -\alpha )}{\sin \alpha .\sin \beta }\]     \[\therefore \,h=a\sin \alpha \sin \beta \,co\text{sec }(\beta -\alpha )\]   and   \[d=h\cot \beta =a\sin \alpha .\cos \beta .\text{cosec}(\beta -\alpha )\]

(2)

\[H=x\cot \alpha \tan (\alpha +\beta )\]            

(3)

\[a=h(\cot \alpha +\cot \beta ),\] where by     \[h=a\sin \alpha .\sin \beta .\text{cosec}(\alpha +\beta )\] and     \[d=h\cot \beta =a\sin \alpha .\cos \beta .\text{cosec }(\alpha +\beta )\]

(4)

\[H=\frac{h\cot \beta }{\cot \alpha }\]                

(5)

\[h=\frac{H\sin (\beta -\alpha )}{\cos \alpha \sin \beta }\]   or \[H=\frac{h\cot \alpha }{\cot \alpha -\cot \beta }\]

(6)

\[H=\frac{a\sin (\alpha +\beta )}{\sin (\beta -\alpha )}\]

(7)

\[AB=CD\]. Then, \[x=y\tan \left( \frac{\alpha +\beta }{2} \right)\]

(8)

     
\[h=\frac{d}{\sqrt{{{\cot }^{2}}\beta +{{\cot }^{2}}\alpha }}\]

(9)  

\[h=\frac{AB}{\sqrt{{{\cot }^{2}}\beta -{{\cot }^{2}}\alpha }}\]

(10)

\[h=AP\sin \alpha \]   \[=a\sin \alpha .\sin \gamma .c\text{osec}(\beta -\gamma )\] and If \[AQ=d\], then \[d=AP\text{ }\cos \alpha =a\text{ }\cos \alpha .\sin \gamma \text{ }.\text{ cosec  }(\beta -\gamma )\]

(11)  

     
\[AP=a\sin \gamma .co\text{sec}(\alpha -\gamma )\],   \[AQ=a\sin \delta .\text{cosec}(\beta -\delta )\]   and apply,   \[P{{Q}^{2}}=A{{P}^{2}}+A{{Q}^{2}}-2AP.AQ\cos \theta \]