On Monday, we posted this challenging ACT trigonometry question, and we know you’ve been waiting impatiently all week for the answer (or at least that’s what we’d like to think). Well all that anticipation wasn’t for nothing! Check out the question and explanation below!

## ACT Challenge Question #15

If sin = – , which of the following could be true about ?

This is a hard problem. First of all, notice that the angle must be in QIII or QIV, since the sine is negative. It has to be a value on the line = – . The first three choices are all in QI and QII, so the angle can’t possibly be in any of those. We can easily eliminate (A) & (B) & (C).

Let’s think about the last two. First, option (D). This option is in QII and QIII, so the part in QIII might go down low enough to contain the angle. The lowest ray in that region is at .

Technically, this ray isn’t even included, because it’s the endpoint of an inequality, but we will just use this value. Sine is positive in QII, zero at the negative x-axis, and starts to get more and more negative as we go around through QIII. How low does this go by this last value?

= – =

This does not go down as far as

Here’s a diagram with region (D) and the line = – .

So, we can eliminate (D).

This leaves (E) by the process of elimination, but let’s verify that this works. Region (E) is in QIV. The sine is negative one at the bottom, where the unit circle intersects the negative y-axis. As we rise from this bottom, we get negative values of smaller and smaller absolute value, until it equals zero at the positive x-axis. We have to know how low the lowest values in this region are. Well, the limiting value is .

We know that = –

Now, if you happen to have the decimal approximations memorized, you will see that:
= –

If you don’t have this decimal equivalent memorized, you could think about the triangles in QIV:

Even if we can’t directly compare the sizes of the two vertical legs, we definitely can compare the two horizontal legs, and 0.8 is definitely bigger than 0.5. That means the purple {0.6, .8, 1} triangle is further around in the counterclockwise direction from the green 30-60-90 triangle, and the point with a sine of – 0.6 would clearly be included in the arc from the 30-60-90 triangle up to the positive x-axis.