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# Solving sinusoidal equations of the form sin(x)=d

Sal finds the expressions that together represent all possible solutions to the equation sin(x)=1/3. Created by Sal Khan.

Video transcript

Voiceover:Which of these are contained in the solution set to
sine of X is equal to 1/3? Answers should be rounded
to the nearest hundredths. Select all that apply. I encourage you to pause
the video right now and work on it on your own. I'm assuming you've given a go at it. Let's think about what this is asking. They're asking what are the X values? What is the solution set? What are the possible X values where sine of X is equal to 1/3? To help us visualize this, let's draw a unit circle. That's my Y axis. This right over here is my X axis. This is X set one. This is Y positive one. Negative one along the X axis, and negative one on the Y axis. The unit circle, I'm going
to center it at zero. It's going to have a radius of one, a radius of one and we just
have to remind ourselves what the unit circle definition
of the sine function is. If we have some angle,
one side of the angle is going to be a ray
along the positive X axis, if we do this in a color you can see, along the positive X axis. Then the other side, so let's see, this is our angle right over here. Let's say that's some angle theta. The sine of this angle is going to be the Y value of where this ray intersects the unit circle. This right over here, that is going to be sine of theta. With that review out of the way, let's think about what X values, and we're assuming we're
dealing in radians. What X values when if
I take the sine of it are going to give me 1/3? When does Y equal 1/3
along the unit circle? That's 2/3, 1/3 right over here. We see it equals 1/3 exactly two places, here and here. There's two angles where, or at least two, if we just take one or two on each pass of the unit circle. Then we can keep adding
multiples of two pi to get as many as we want. We see just on the unit circle
we could have this angle. We could have this angle right over here. Or we could go all the way around to that angle right over there. Then we could add any multiple of two pi to those angles to get other angles that would also work where if I took the sine of them I would get 1/3. Now let's think about what these are. Here we can take our calculator out, and we could take the inverse sine of 1/3. Let's do that. The inverse sine of 1 over 3. We have to remember what the range of the inverse sine function is. It's going to give us a value between negative pi over 2 and pi over 2, so a value that sticks us in either the first or the fourth quadrant if we're thinking about the unit
circle right over here. We see that gave us zero point, if we round to the nearest hundredth, 34. Essentially they've given us this value. They've given us 0.34. That's this angle right over here. How did I know that? Well, it's a positive value. It's greater than zero, but it's less than pi over two. Pi is 3.14, so pi over two is going to be 1.57 and we can go on and on and on so this right over here is 0.34 radians. But what would this thing over here be? It's going to be whatever, if we go to the negative X axis and we subtract 0.34 so we subtracted 0.34. This is 0.34 We're going to get to this angle. It's going to be if we take pi minus the previous answer, it gets us we round to the nearest hundredth, is 2.8 radians. This is 0.34 radians, and then this one, let me do it in this purple color, this one right over here if we were to go all the way around, it's pi minus 0.34 which is 2.80 radians rounding to the nearest hundredth. Now that's not all of the values. We can add multiples of
two pi to each of these. So 2.80 plus any multiple of two pi's, so two pi N where N is an integer. N is an integer. Or we could take 0.34 and add any multiple of two pi. So two pi N where N is an integer. Our solution set here
just to rewrite it kind of outside of this messiness is going to be 2.80 radians plus two pi N where N is an integer, and 0.34 plus two pi N
where N is an integer. Let's see which of these are at least a subset of this. We look at 0.34 plus two
pi N where N is an integer. That's exactly what we wrote over here. That's 0.34. If N is a positive integer
we'll go around this way and we keep getting
back to the same point. If it's a negative integer
we go around that way. We keep getting to the same point, but that's definitely in the solution set. 0.34 plus pi N for N an integer. So if we have 0.34 and
if we were to not add two pi, but just pi where would we get to? Well, we would get to right over there. The sine of this isn't
going to be positive 1/3. It's going to be negative 1/3. So we could rule that out. Negative 0.34, well that's
this angle right over here. The sine of that's going
to be negative 1/3. If you add a multiple of two pi to that, you're still going to get negative 1/3 so that doesn't work. Same thing for this one right over here. 2.8 plus two pi N, that's
what we wrote right over here. 2 point going all the way to 2.8 and any multiple of it
is going to get you back to that same point so that one works. 2.8 plus pi N so if you're here and if you added pi you're
going to get over here, and the sine of that isn't
going to be positive 1/3. It's going to be negative 1/3 so we can rule this one out as well. These are the only two apply, and if you actually take them together you have the entire solution set to this equation right over here.