Trigonometry

1. Sine formula :a/sin A = b/sin B = c/sin C = rTherefore, sin A = a/r, sin B = b/r, sin C = c/r
sin^2 A + sin^2 B = sin^2 C==> (a/r)^2 + (b/r)^2 = (c/r)^2==> a^2 + b^2 = c^2Using the converse of Pythagoras Theorem, triangle ABC is a right-angled with right angle C.
2. Cosine formula :cos A = (b^2 + c^2 - a^2)/(2bc)cos B = (c^2 + a^2 - b^2)/(2ca)cos C = (a^2 + b^2 - c^2)/(2ab)
a cos A + b cos B = c cos C==> a(b^2 + c^2 - a^2)/(2bc) + b(c^2 + a^2 - b^2)/(2ca) = c(a^2 + b^2 - c^2)/(2ab)==> a^2(b^2 + c^2 - a^2) + b^2(c^2 + a^2 - b^2) = c^2(a^2 + b^2 - c^2)==> a^2b^2 + a^2c^2 - a^4 + b^2c^2 + a^2b^2 - b^4 = a^2c^2 + b^2c^2 - c^4==> 2a^2b^2 - a^4 - b^4 + c^4 = 0==> c^4 = a^4 - 2a^2b^2 + b^4==> c^4 = (a^2 - b^2)^2==> c^2 = a^2 - b^2 or c^2 = b^2 - a^2==> b^2 + c^2 = a^2 or c^2 + a^2 = b^2
Using the converse of Pythagoras Theorem,
triangle ABC is a right-angled with right angle A or B.

(sinA)^2+(sin B)^2=(sinC)^2
(sinA)^2+(sin B)^2=(sin(A+B))^2
(sinA)^2+(sin B)^2=(sinAcosB+sinBcosA)^2
(sinA)^2+(sin B)^2=((sinA)^2)(cosB)^2+(cosA)^2(sinB)^2+2sinAcosAsinBcosB
2((sinA)^2)(sinB)^2=2sinAcosAsinBcosB
sinAsinB(cosAcosB-sinAsinB)=0
sinAsinBcos(A+B)=0
As A and B are not equal to zero,
cos(A+B)=0
A+B=90
180-C=90
C=90