Question

Please help!!! Medal and fan will be given
Find all solutions in the interval [0, 2π).
sin2 x + sin x = 0

Answer 1

$$\sin^2x+\sin x=0\\\sin x\cdot(\sin x+ 1)=0$$by the fact that \(ab=0\) suggests either \(a=0\) or \(b=0\) we can conclude that either \(\sin x=0\) or \(\sin x + 1=0\)

Answer 2

for \(\sin x=0,x\in[0,2\pi)\) we know that \(x\in\{0,\pi\}\) (look at the unit circle if you need convincing)

Answer 3

now we focus on \(\sin x+1=0\) i.e. \(\sin x=-1\). there is precisely one point on the unit circle with a \(y\)-coordinate of \(-1\):