Interpreting solutions of trigonometric equations | Trigonometry | Precalculus | Khan Academy

Alvaro presses the treadle of a spinning wheel with his foot. It moves a bar up and down, making the wheel spin. So just to be clear, what a treadle is: this is an old spinning wheel, and this little pedal is a treadle. As this goes up and down, it's going to pull on this bar, which is then going to spin this wheel, which can then be used to essentially power the machine.

So it says the function b of t models the height in centimeters of the top of the bar when Alvaro has pressed the treadle for t seconds. So it's telling us the height. I can barely see where the top of the bar is; someplace over here. This isn't exactly what they're probably talking about in this exercise here, but this is just to give you a visualization of what a treadle is, and what the bar is, and then what the spinning wheel is.

Alvaro has pressed that treadle for t seconds, so they give us b of t right over here: 90 minus 12 times sine of 5t. The first question is, what does the solution set to y equal to 90 minus 12 times sine of 5 times 6 represent? Pause this video and see if you can think about that.

All right, so it looks like right over here. So we have the 90, 90, 12, 12, and we're subtracting 12 sine of 5 times t, 5 times t. So this right over here is t. So this is the solution. The solution set right over here tells us what is the height because that's what b of t is. So b of t is equal to y. What is the height when t is equal to six? Remember, t is in seconds, so this is height—height of the top of the bar at 6 seconds.

All right, now we have more questions here. The next question asks us, what does the solution set to 95 equals 90 minus 12 sine of 5t represent? Pause the video and think about that.

All right, so here they're saying that b of t is equal to 95, and so the solution set—you're really solving for t. So you're really solving for all of the times when our height is going to be 95 centimeters. So all times t when height of the top of the bar is at 95 centimeters. That's going to keep happening over and over and over again as t goes forward in time.

So you're going to have a very large—you're going to have an infinite solution set over here. You're going to have an infinite number of t's at which your solution, at which the top of the bar is at 95 centimeters.

Now we have another question. This one is asking us, what does the solution set to y equal to 90 minus 12 sine of pi over 2 represent? So pause the video and think about that.

All right, now this is pretty interesting; we can actually evaluate what sine of pi over 2 is. So sine of pi over 2 radians or sine of 90 degrees, that is going to be equal to 1. And so that's the maximum value that this sine over here can take on.

Now we're going to subtract 12 times that. So this is taking on a max. Then, when you subtract 12 times that, this is actually the minimum value that you can take on. You're going to have—you can't get any lower than this. And so this is going to be the lowest—the lowest height for the top of the bar.

And we're done.