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Trig word problem: solving for temperature

Video transcript

Voiceover:In the last video, we were able to model the average high temperature in Santiago, Chile as a function of days as we go through the year, where the days were the days after January 7th, so this right over here. Day zero is actually January 7th. But we weren't done. They want us to figure out how many days after January 7th is the first spring day, when the temperature reaches 20 degrees celsius And I told you, be careful. Pay attention to this whole notion of the first spring day and the reason is because there's actually two days where the temperature reaches 20 degrees celsius, So let's say this is 20 degrees celsius right over here, notice you have this day right over here and then you have this day right over there and which one is in the spring or the first spring day? Well, if this is - we're in summer right over here We're in the southern hemisphere, so our summer is going to be when it is the winter in the northern hemisphere. Summer, what season comes after summer? Well, this is going to be the fall, this is going to be the winter, Now this is going to be the spring. Of course, you go back to summer. So, we want this value, not this value. This one will be the day in fall when the average high temperature is 20 degrees celsius, This is the first day of spring where the average high temperature is 20 degrees celsius. The first spring day. I guess it's not necessarily the first spring day, it's the first spring day where the temperature reaches 20 degrees celsius. So, this is happening in the spring, so this is the value we want. So, let's just think about that. As we try to manipulate this a little bit. So, we want to get to 20 degrees celsius, so we could write 20 is equal to 7.5 times cosine of 2 pi over 365 times the days, plus 21.5 Now, we could subtract 21.5 from both sides and we get -1.5 is equal to- and I'll just copy and paste all of this- is going to be equal to that. So, copy and paste. It's going to be equal to that right over there. Now I could divide both sides by 7.5. Notice, I'm trying to solve for cosine, and eventually solve for D, but we're going to take a pause, once we have this in terms of cosine, we have to be careful here. So we're going to divide both sides by 7.5 And we're going to get, let's see... Actually, I don't even need a calculator for this.. 1.5 divided by 7 - this is one fifth. 5 times 15, or 5 times 1.5 is 7.5 so this is -1/5 or I could write it as -0.2 is equal to cosine of 2 pi over 365 times days after January 7th. Now, this is where we have to be very careful instead of just blindly applying the inverse cosine function, we have to make sure which angle we are actually getting. That we're getting the right angle over here. 'Cause remember, we want the argument to the cosine, that doesn't give us this point, that gives us this point right here. Or that corresponds to this point right over there. So let's draw a unit circle, just to make sure we know what's going on. And I actually do this all the time, especially if I'm trying to apply the inverse trigonometric functions in a kind of applied context, where I just can't blindly plug it in to my calculator. So, let me draw a unit circle right over here. X-axis, y-axis... Circle of radius 1, centered at the origin You get the picture we've done this many times before. And January 7th, that corresponds to this point right over here. That's that point right over here, that we are in the summer. Then, as the days go by, our argument to the cosine increases, the angle increases, and this right over here will be the fall. So, this point right over here, so we're at the fall right over here. And then we move on to the winter right over there. And then finally, we go to the spring. This is the spring right over there. And we want the angle that gets us -0.2 So, this is -1. -0.2 is a fifth of the way, so it's -0.2 and notice, there's two angles that get us there. There's this angle, there's that angle right over there. And then there's also, let me draw a little dotted line here, there's that angle. But then there's also this angle. which is going even further. Or another way you could think about it, you could go backwards to get to that angle and then if you wanted to go all the way around, to the next spring, you could add 2 pi to it. So which one do we want? Of course, we want the one in the spring. But if we just blindly apply the inverse cosine, of -0.2, that's going to give us this one. And we can verify that. So, let's see... Inverse cosine of -0.2 is 1.77. Remember, this is zero, this is pi right over, so this is 3.14159, and this is 1.77, so notice it's a little bit more than half of pi which is exactly what it gave- it gave this angle right over here. So this is approximately 1.77 radiants is this angle, is that angle right over there. But we don't want that one, we want this one, so how do we figure that out? Well, we could view it as, we could go all the way around to 2 pi, and then subtract 1.77, so we could say 2 pi minus 1.77 roughly to get this angle, so let's do that. Let's take 2 times pi, and then subtract this number and I'm going to do second answer, so answer is just the previous answer that I got so that way I have good precision here. So I get 4.511, and we can kind of make sure that is the right thing because that's going to be between pi and 2 pi. 2 pi is 2 times 3.14159, so it's going to be 6.28 something. This is 3.14159, this is the right angle. Now, we're not done yet. That's just the angle, that's the argument that we need to give in here to get to that point, but what's the days going to be. Well 2 pi over 365 days is going to be equal to this number 4.511, so let me write that down. So, this right over here is going to be approximately equal to 4.511. So, we scroll down a little bit We can say 2 pi over 365 times the days after January 7th is approximately going to be equal to 4.511 to solve for days, we can just multiply both sides by the reciprocal of the coefficient, so we're going to multiply both sides by 365 over 2 pi. So that's going to cancel, and now we can use our calculator for this. So, we've got much better precision there. So let's take our previous answer, times 365, we deserve a drumroll here... Divided by 2 pi, and we get if we round to the nearest day 262 days after January 7th. So 262 days. and we are done.