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=π−C
Taking the sine on both sides of the equation:
sin(A+B)=sin(π−C)
sinAcosB+cosAsinB=sinπcosC−cosπsinC
sinAcosB+cosAsinB=(0)(cosC)−(−1)(sinC)
sinAcosB+cosAsinB=sinC---(1)
Taking the cosine on both sides of the equation:
cos(A+B)=cos(π−C)
cosAcosB−sinAsinB=cosπcosC+sinπsinC
cosAcosB−sinAsinB=(−1)(cosC)+(0)(sinC)
cosAcosB−sinAsinB=−cosC---(2)
Dividing (2) by (1), we see that we have:
cosAcosB−sinAsinBsinAcosB+cosAsinB=−cosCsinC
Dividing the numerator and denominator on the left by sinAsinB, we get:
cosAcosBsinAsinB−sinAsinBsinAsinBsinAcosBsinAsinB+cosAsinBsinAsinB=−cotC
cotAcotB−1cotB+cotA=−cotC
cotAcotB−1=(−cotC)(cotB+cotA)
cotAcotB−1=−cotCcotB−cotCcotA)
cotAcotB+cotBcotC+cotAcotC=1
2.
A+B+C=π
A2+B2=π2−C2
Taking the sine on both sides of the equation:
sin(A2+B2)=sin(π2−C2)
sinA2cosB2+cosA2sinB2=sinπ2cosC2−cosπ2sinC2
sinA2cosB2+cosA2sinB2=(1)cosC2−(0)sinC2
sinA2cosB2+cosA2sinB2=cosC2---(3)
Taking the cosine on both sides of the equation:
cos(A2+B2)=cos(π2−C2)
cosA2cosB2−sinA2sinB2=cosπ2cosC2+sinπ2sinC2
cosA2cosB2−sinA2sinB2=(0)cosC2+(1)sinC2
cosA2cosB2−sinA2sinB2=sinC2---(4)
Dividing (3) by (4), we see that we have:
sinA2cosB2+cosA2sinB2cosA2cosB2−sinA2sinB2=cosC2sinC2
sinA2cosB2+cosA2sinB2cosA2cosB2−sinA2sinB2=cotC2
Dividing the numerator and denominator on the left by sinA2sinB2, we get:
sinA2cosB2sinA2sinB2+cosA2sinB2sinA2sinB2cosA2cosB2sinA2sinB2−sinA2sinB2sinA2sinB2=cotC2
cotB2+cotA2cotA2cotB2−1=cotC2
cotB2+cotA2=cotA2cotB2cotC2−cotC2
cotA2+cotB2+cotC2=cotA2cotB2cotC2
No comments:
Post a Comment