From where does this - ve sign come from?

Question

Let

θ=54

5θ=270

2θ+3θ=270

2θ=270-3θ

sin(2θ)=sin(270-3θ)

sin(2θ)= -cos(3θ)

From where does - ve sign come from?

.

270-3θ will be in 2quad where sin is +ve.

Answer 1

sin(A-B) = sinAcosB - cosAsinB

So, sin(270-3θ) = sin270cos(3θ) - cos270sin(3θ)

= (-1)cos(3θ) - (0)sin(3θ) = -cos(3θ)