Solving $\sin(-\theta)=0.35 $. Is $\sin$ postive or negative? Where are the angles located?

Question

My question:
$$\sin(-\theta)=0.35 \qquad\text{range: } 0<\theta<360$$

Is $sin$ positive or negative in this case? and where would the angles locate at?

Thank you!

Answer 1

In these type of question, we should draw the graph of $\sin x$, and do some algebraic manipulation based on that. We know that $$\sin(-\theta) = -\sin(\theta)$$ therefore we are trying to find the solution of $\sin x = -0.35,$ where $x \in (0^{\circ},360^{\circ})$.

From the graph, we see that, $\sin(x)$ is negative in $x \in(180^{\circ},360^{\circ})$, hence we will have two solutions one in each of the intervals $(180^{\circ},270^{\circ})$ and $(270^{\circ},360^{\circ})$ (as it takes all values between $0$ and $-1$ in both of these intervals).

If you want better bounds on where the roots will lie, you should consider the increasing-decreasing nature of the graph and decimal values of the known values of sin function (For example, $\sin(x)$ takes values between $0$ and $-0.707$ for $x \in (180^{\circ},225^{\circ}),$ so $-0.35$ will occur in this interval between $180^{\circ}$ and $270^{\circ},$ which is a better bound).

Hope it helps:)