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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)

cotAcotB+cotBcotC+cotAcotC=1

2.


A+B+C=π

A2+B2=π2C2

Taking the sine on both sides of the equation:

sin(A2+B2)=sin(π2C2)

sinA2cosB2+cosA2sinB2=sinπ2cosC2cosπ2sinC2

sinA2cosB2+cosA2sinB2=(1)cosC2(0)sinC2

sinA2cosB2+cosA2sinB2=cosC2---(3)

Taking the cosine on both sides of the equation:

cos(A2+B2)=cos(π2C2)

cosA2cosB2sinA2sinB2=cosπ2cosC2+sinπ2sinC2

cosA2cosB2sinA2sinB2=(0)cosC2+(1)sinC2

cosA2cosB2sinA2sinB2=sinC2---(4)

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

sinA2cosB2+cosA2sinB2cosA2cosB2sinA2sinB2=cosC2sinC2

sinA2cosB2+cosA2sinB2cosA2cosB2sinA2sinB2=cotC2

Dividing the numerator and denominator on the left by sinA2sinB2, we get:

sinA2cosB2sinA2sinB2+cosA2sinB2sinA2sinB2cosA2cosB2sinA2sinB2sinA2sinB2sinA2sinB2=cotC2

cotB2+cotA2cotA2cotB21=cotC2

cotB2+cotA2=cotA2cotB2cotC2cotC2

cotA2+cotB2+cotC2=cotA2cotB2cotC2

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