Menu Close

nice-calculus-in-AB-C-sin-2-A-sin-2-B-sin-2-C-2-prove-that-AB-C-is-right-triangle-Good-luck-




Question Number 124501 by mnjuly1970 last updated on 03/Dec/20
          ....nice  calculus..     in AB^Δ C : sin^2 (A)+sin^2 (B)+sin^2 (C)=2  prove that: AB^Δ C is right triangle                 Good luck.
.nicecalculus..inABCΔ:sin2(A)+sin2(B)+sin2(C)=2provethat:ABCΔisrighttriangleGoodluck.
Answered by mindispower last updated on 03/Dec/20
sin(c)=sin(a+b)  ⇔sin^2 (A)+sin^2 (B)+(sin(A+B))^2 )=2  sin^2 (A+B)=sin^2 (A)cos^2 (B)+cos^2 (A)sin^2 (B)  +(1/2)sin(2A)sin(2B)  sin^2 (A)−1+sin^2 (B)−1+((sin(2A)sin(2B))/2)+  sin^2 (A)cos^2 (B)+sin^2 (B)cos^2 (A)=0  ⇔−cos^2 (A)+cos^2 (A)sin^2 (B)−cos^2 (B)+cos^2 (B)sin^2 (A)  +((sin(2A)sin(2B))/2)=0  ⇔−2cos^2 (A)cos^2 (B)+2sin(A)sin(B)cos(A)cos(B)=0  ⇔2cos(A)cos(B)(sin(A)sin(B)−cos(A)cos(B))=0  ⇒cos(A)cos(B)cos(A+B)=0  ⇒A=(π/2),B=(π/2),or A+B=(π/2)⇒C=(π/2)
sin(c)=sin(a+b)sin2(A)+sin2(B)+(sin(A+B))2)=2sin2(A+B)=sin2(A)cos2(B)+cos2(A)sin2(B)+12sin(2A)sin(2B)sin2(A)1+sin2(B)1+sin(2A)sin(2B)2+sin2(A)cos2(B)+sin2(B)cos2(A)=0cos2(A)+cos2(A)sin2(B)cos2(B)+cos2(B)sin2(A)+sin(2A)sin(2B)2=02cos2(A)cos2(B)+2sin(A)sin(B)cos(A)cos(B)=02cos(A)cos(B)(sin(A)sin(B)cos(A)cos(B))=0cos(A)cos(B)cos(A+B)=0A=π2,B=π2,orA+B=π2C=π2
Commented by mnjuly1970 last updated on 03/Dec/20
thank you so much  mr mindspower  nice as always...
thankyousomuchmrmindspowerniceasalways
Answered by Dwaipayan Shikari last updated on 03/Dec/20
2sin^2 A+2sin^2 B+2sin^2 C=4  1−cos2A+1−cos2B+1−sin2C=4  cos2A+cos2B+cos2C=−1  2cos(A+B)cos(A−B)+cos2C=−1  2cos(A+B)cos(A−B)+2cos^2 C=0  cos(A+B)cos(A−B)+cos^2 C=0  cosC cos(A−B)=cos^2 C  cosC(cosC−cos(A−B))=0  cosC=0  C=(π/2)      or  cosC=cos(A−B) ⇒C=A−B⇒C+B=A
2sin2A+2sin2B+2sin2C=41cos2A+1cos2B+1sin2C=4cos2A+cos2B+cos2C=12cos(A+B)cos(AB)+cos2C=12cos(A+B)cos(AB)+2cos2C=0cos(A+B)cos(AB)+cos2C=0cosCcos(AB)=cos2CcosC(cosCcos(AB))=0cosC=0C=π2orcosC=cos(AB)C=ABC+B=A
Commented by mnjuly1970 last updated on 03/Dec/20
thank you so much  extraordinary...
thankyousomuchextraordinary

Leave a Reply

Your email address will not be published. Required fields are marked *