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Thursday, January 7, 2016

What else could we generate from A+B+C=π, when A,B,C are three angles from a triangle?

Following previous blog post, we have shown that if

If A,B,C are three angles from a triangle, i.e. A+B+C=π, then we should have known the following equality holds.

tanA+tanB+tanC=tanAtanBtanC

On this blog post, we now try to generate another relation between A,B and C for cotangent functions:

1.

A+B+C=π

A+B=πC

Taking the sine on both sides of the equation:

sin(A+B)=sin(πC)

sinAcosB+cosAsinB=sinπcosCcosπsinC

sinAcosB+cosAsinB=(0)(cosC)(1)(sinC)

sinAcosB+cosAsinB=sinC---(1)

Taking the cosine on both sides of the equation:

cos(A+B)=cos(πC)

cosAcosBsinAsinB=cosπcosC+sinπsinC

cosAcosBsinAsinB=(1)(cosC)+(0)(sinC)

cosAcosBsinAsinB=cosC---(2)

Dividing (2) by (1), we see that we have:

cosAcosBsinAsinBsinAcosB+cosAsinB=cosCsinC

Dividing the numerator and denominator on the left by sinAsinB, we get:

cosAcosBsinAsinBsinAsinBsinAsinBsinAcosBsinAsinB+cosAsinBsinAsinB=cotC

cotAcotB1cotB+cotA=cotC

cotAcotB1=(cotC)(cotB+cotA)

cotAcotB1=cotCcotBcotCcotA)

\displaystyle \color{yellow}\bbox[5px,green]{\cot A \cot B+\cot B\cot C+\cot A\cot C=1}

2.


A+B+C=\pi

\displaystyle \frac{A}{2}+\frac{B}{2}=\frac{\pi}{2}-\frac{C}{2}

Taking the sine on both sides of the equation:

\displaystyle \sin \left(\frac{A}{2}+\frac{B}{2}\right)=\sin \left(\frac{\pi}{2}-\frac{C}{2}\right)

\displaystyle \sin \frac{A}{2} \cos \frac{B}{2}+\cos \frac{A}{2}\sin \frac{B}{2}=\sin\frac{\pi}{2} \cos \frac{C}{2}-\cos \frac{\pi}{2} \sin \frac{C}{2}

\displaystyle \sin \frac{A}{2} \cos \frac{B}{2}+\cos \frac{A}{2}\sin \frac{B}{2}=(1)\cos \frac{C}{2}-(0)\sin \frac{C}{2}

\displaystyle \sin \frac{A}{2} \cos \frac{B}{2}+\cos \frac{A}{2}\sin \frac{B}{2}=\cos \frac{C}{2}---(3)

Taking the cosine on both sides of the equation:

\displaystyle \cos \left(\frac{A}{2}+\frac{B}{2}\right)=\cos \left(\frac{\pi}{2}-\frac{C}{2}\right)

\displaystyle \cos \frac{A}{2} \cos \frac{B}{2}-\sin \frac{A}{2}\sin \frac{B}{2}=\cos \frac{\pi}{2} \cos \frac{C}{2}+\sin \frac{\pi}{2} \sin \frac{C}{2}

\displaystyle \cos \frac{A}{2} \cos \frac{B}{2}-\sin \frac{A}{2}\sin \frac{B}{2}=(0) \cos \frac{C}{2}+(1) \sin \frac{C}{2}

\displaystyle \cos \frac{A}{2} \cos \frac{B}{2}-\sin \frac{A}{2}\sin \frac{B}{2}= \sin \frac{C}{2}---(4)

Dividing (3) by (4), we see that we have:

\displaystyle \frac{\sin \frac{A}{2} \cos \frac{B}{2}+\cos \frac{A}{2}\sin \frac{B}{2}}{\cos \frac{A}{2} \cos \frac{B}{2}-\sin \frac{A}{2}\sin \frac{B}{2}}=\frac{\cos \frac{C}{2}}{\sin \frac{C}{2}}

\displaystyle \frac{\sin \frac{A}{2} \cos \frac{B}{2}+\cos \frac{A}{2}\sin \frac{B}{2}}{\cos \frac{A}{2} \cos \frac{B}{2}-\sin \frac{A}{2}\sin \frac{B}{2}}=\cot \frac{C}{2}

Dividing the numerator and denominator on the left by \sin \frac{A}{2}\sin \frac{B}{2}, we get:

\displaystyle \frac{\frac{\sin \frac{A}{2} \cos \frac{B}{2}}{\sin \frac{A}{2}\sin \frac{B}{2}}+\frac{cos \frac{A}{2}\sin \frac{B}{2}}{\sin \frac{A}{2}\sin \frac{B}{2}}}{\frac{\cos \frac{A}{2} \cos \frac{B}{2}}{\sin \frac{A}{2}\sin \frac{B}{2}}-\frac{\sin \frac{A}{2}\sin \frac{B}{2}}{\sin \frac{A}{2}\sin \frac{B}{2}}}=\cot \frac{C}{2}

\displaystyle \frac{\cot \frac{B}{2}+\cot \frac{A}{2}}{\cot \frac{A}{2} \cot \frac{B}{2}-1}=\cot \frac{C}{2}

\displaystyle \cot \frac{B}{2}+\cot \frac{A}{2}=\cot \frac{A}{2} \cot \frac{B}{2}\cot \frac{C}{2} -\cot \frac{C}{2}

\displaystyle \color{yellow}\bbox[5px,purple]{\cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}=\cot \frac{A}{2} \cot \frac{B}{2}\cot \frac{C}{2} }

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