If $\sin X$ and $\cos Y$ are both negative, what can we say about $X + Y$?

If $\sin X$ and $\cos Y$ are both negative, what can we say about $X + Y$?

Question

1. Given that $\sin 3z^\circ = k$, where $0 < k < 1$, how many values of $z$ lie between $0$ and $360$?

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1. Given that $X$ and $Y$ both lie between $0$ and $360$, and that $\sin X^\circ$ and $\cos Y^\circ$ are both negative, find the values between which $X + Y$ must lie.

Suggestion

How does $\sin 3z^\circ$ relate to the function $\sin z^\circ$?

Could drawing a graph of $y = \sin (3z)$ help us here?

Solution

The function $y=\sin 3z$ is a stretch of $y=\sin z\:$, scale factor $\dfrac{1}{3}$, in the $x$-direction.

Thus, instead of one ‘complete wave’ between $0$ and $360$, we have three, giving this graph:

So while we would have two intersections between $y = k$ and $y = \sin z$ for $0 < z < 360$, we’ll instead have six in the same interval.

If $\sin X< 0$ and $0 < X < 360$, then $180 < X < 360$, while if $\cos Y< 0$ and $0 < Y < 360$, then $90 < Y < 270$.

So for $\sin X$ and $\cos Y$ to both be negative, $270 < X+Y < 630$.

Note that if $270 < X+Y < 630$, this is no guarantee that $\sin X$ and $\cos Y$ are both negative (for example, $X = 170, Y = 110$).