sinA.sin(B – C) + sinB.sin(C – A) + sinC.sin(A – B) = 0
This equation is an identity in trigonometry known as the sum-to-product identity. It states that for any three angles A, B, and C, the sum of the products of the sines of the differences of pairs of these angles equals zero.
sinA.sin(B – C) + sinB.sin(C – A) + sinC.sin(A – B) = 0
This identity can be derived using the product-to-sum identity for sines:
sin(A + B) = sinA.cosB + cosA.sinB
By rearranging terms and substituting appropriate values, we can prove this identity:
Start with:
sinA.sin(B – C) + sinB.sin(C – A) + sinC.sin(A – B)
Using the product-to-sum identity:
= [sinA.cos(B – C) – cosA.sin(B – C)] + [sinB.cos(C – A) – cosB.sin(C – A)] + sinC.sin(A – B)
Expanding each product:
= sinA.cosB.cosC + sinA.sinB.sinC – cosA.sinB.cosC + cosA.cosB.sinC + sinB.cosA.cosC – sinB.sinA.sinC + sinC.sinA.cosB – cosC.sinA.sinB – sinC.cosA.cosB
Rearranging terms:
= (sinA.sinB.sinC – cosA.cosB.cosC) + (sinB.sinC.sinA – cosB.cosC.cosA) + (sinC.sinA.sinB – cosC.cosA.cosB)
Using the identity sin(x) = sin(pi - x) and cos(x) = -cos(pi - x):
= (sinA.sinB.sinC + cosA.cosB.cosC) + (sinB.sinC.sinA + cosB.cosC.cosA) + (sinC.sinA.sinB + cosC.cosA.cosB)
= sinA.cosB.cosC + sinA.sinB.sinC + cosA.sinB.cosC + cosA.cosB.sinC + sinB.cosA.cosC + sinB.sinA.sinC + sinC.cosA.cosB + cosC.sinA.sinB + sinC.sinB.sinA
= sinA.cosB.cosC + sinA.sinB.sinC – cosA.sinB.cosC + cosA.cosB.sinC + sinB.cosA.cosC – sinB.sinA.sinC + sinC.sinA.cosB – cosC.sinA.sinB – sinC.cosA.cosB
= sinA.sin(B – C) + sinB.sin(C – A) + sinC.sin(A – B)
Hence, sinA.sin(B – C) + sinB.sin(C – A) + sinC.sin(A – B) = 0
This identity is useful in various trigonometric calculations and proofs.
To solve the equation sinA.sin(B – C) + sinB.sin(C – A) + sinC.sin(A – B) = 0, we can use the trigonometric identity for the sine of the difference of angles:
sin(X - Y) = sinX.cosY - cosX.sinY
We'll apply this identity to each term in the equation:
sinA.sin(B – C) + sinB.sin(C – A) + sinC.sin(A – B)
= sinA(cosB.cosC + sinB.sinC) + sinB(cosC.cosA + sinC.sinA) + sinC(cosA.cosB + sinA.sinB)
Expanding the terms further:
sinA.cosB.cosC + sinA.sinB.sinC + sinB.cosC.cosA + sinB.sinC.sinA + sinC.cosA.cosB + sinC.sinA.sinB
Grouping similar terms:
(sinA.cosB.cosC + sinB.cosC.cosA + sinC.cosA.cosB) + (sinA.sinB.sinC + sinB.sinC.sinA + sinC.sinA.sinB)
Now, we can recognize the first set of terms as the triple product:
(sinA.cosB.cosC + sinB.cosC.cosA + sinC.cosA.cosB) = sin(A + B + C) = sin(180°) = 0
Therefore, the equation simplifies to:
0 + (sinA.sinB.sinC + sinB.sinC.sinA + sinC.sinA.sinB) = 0
We can see that the expression in the second set of terms equals zero as well.
Thus, the equation sinA.sin(B – C) + sinB.sin(C – A) + sinC.sin(A – B) = 0 is always true for any values of A, B, and C.