IF A+B+C=Π prove that sin(3A)+sin(3B)+sin(3C)=-4cos(3A÷2)cos(3B÷2)cos(3C÷2)

You should start here, and it should be easy to adjust to using 3A, 3B, 3C.

https://socratic.org/questions/if-a-b-c-pi-then-prove-that-sina-sinb-sinc-4cos-a-2-cos-b-2-cos-c-2

Did you know?

Did you know that in trigonometry, there is a fascinating relationship between the sums of angles and various trigonometric functions? Let's consider a scenario where we have three angles, A, B, and C, and their sum is equal to the mathematical constant π (pi). Now, an intriguing mathematical proof shows that if A + B + C = π, then the sum of the sines of three times each angle (3A, 3B, and 3C) is equivalent to negative four times the product of the cosine of half of each angle (3A/2, 3B/2, and 3C/2). In simple terms, sin(3A) + sin(3B) + sin(3C) = -4cos(3A/2)cos(3B/2)cos(3C/2). This relationship highlights the interconnected nature of trigonometric functions, allowing us to derive new information by manipulating existing formulas. It showcases the elegance and beauty of mathematics, where seemingly diverse concepts come together and form meaningful connections.