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Interpreting solutions of trigonometric equations

Starting from a context represented by a trigonometric function, interpret equations based on the function. Created by Sal Khan.

Video transcript

- [Instructor] Alvaro presses the treadle of a spinning wheel with his foot. It moves a bar up and down making the wheels spin. So just to be clear what a treadle is this is an old spinning wheel and this little pedal that is a treadle. And as this goes up and down, it's gonna 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 of, I can barely see where the top of the bar is someplace over here. And 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 those treadle for t seconds. So they give us B of t right over here and 90 minus 12 times sine of 5t. The first question is, what does the solution set to y is equal to 90 minus 12 times sine of five times six, 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 five times t, five times t. So this right over here is t. 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? And remember, t is in seconds. So, this is height, height of top of bar, top of bar at six 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 sign 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 top of bar, of top of bar at 95 centimeters. And 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 gonna have an infinite solution set over here. You're gonna 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 is equal to 90 minus 12 sine of pi over two 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 two is. So sign of pi over two radians or sign of 90 degrees that is going to be equal to one. And so, that's the maximum value that this sign 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 gonna 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.