I don't understand the following example from Math book.
Solve for the equation sin(theta) = -0.428 for theta in radians to 2 decimal places. where 0<= theta<= 2PI.
And this is the answer:
theta=-0.44 + 2PI = 5.84rad and theta = PI-(0.44) = 3.58rad
I don't understand the part why we need to add 2PI in the first answer and add PI in second answer?
On
The sine function is periodic, so once you find that $\theta=-0.44$ satisfies the equation, so does $-0.44+2k\pi$. Here you need to take $k=1$ to get the solution in the range $0 \le \theta \lt 2\pi$. The second comes from the identity $\sin(\pi-x)=\sin(x)$. You lost a minus sign in the question. Now you are in range already so don't need to add any $2\pi$s.
Geometrically, the solutions are in the third and fourth quadrant. The first number generated by a computer is not even in the range $[0,2\pi]$, but it geometrically corresponds to the fourth quadrant. So you can get the fourth quadrant answer by adding $2\pi$ to it. To get the third quadrant answer, you reflect the previous answer through the $y$ axis. The fourth quadrant answer was $0.44$ radians below the $x$-axis, so the reflection is too, which makes it $\pi+0.44$ radians.