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# Trig word problem: solving for temperature

## Video transcript

in the last video we were able to map and model the average high temperature in Santiago Chile is a function of days as we go through the year where the days were the days after January 7 so this right over here de 0 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 I had to 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 that 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 that we're in summer right over here we're in summer right over here when the we're in the southern hemisphere so our summer is going to be when it's the winter in the northern hemisphere is sommore 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 and then of course you go back to the summer so we want this value not this value this one will be the day and fall when the average high temperatures 20 degrees Celsius this is the first day of spring where the average high temperature is 20 degrees Celsius the first spring day oh I guess it's not necessarily the first day of spring it's the first spring day when the temperature reaches 20 degrees Celsius so this is happening this is happening in the spring right earlier so this is the value that 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 20 degrees Celsius so we could write 20 is equal to is equal to 7.5 times cosine cosine of 2 pi over 365 times the days times the days plus 20 1.5 plus twenty 1.5 now we could subtract 21.5 from both sides and we get negative 1.5 is equal to and I'll just copy and paste all of this it's 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 little pause once we have this in terms of cosine it would 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 point 5 this is 1/5 5 times 15 is or 5 times 1.5 is 7.5 so this is negative 1/5 or I could write it as negative 1/5 or I could write it as negative zero point 2 is equal to is equal to cosine of 2 pi over three hundred and sixty five times days after January 7th times days after January seventh now this is where we have to be very very careful instead of just blindly applying the inverse cosine function have to make sure which which angle we are actually getting that we're getting the right angle right over here to 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 let's draw a unit circle just to just to make sure we know what's going on I actually do this all the time especially if I'm trying to apply the inverse inverse trigonometric functions in kind of a applied context where I just can't blindly blindly plug it in into my calculator so let me draw a unit circle right over here x-axis y-axis so circle of radius 1 centered at the origin so that looks looks well you get the picture we've done this many times before and January seventh that corresponds to this point right over here that's that point right over here that we are in the summer 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 the winter right over there and then finally finally we go to the spring this is the spring right over there and we want the angle that gets us negative 0.2 so this is negative 1 negative 1 negative 0.2 fifth of the way so it's negative zero point two and notice there's two angles that get us there there's this angle and there's that angle right over there and then there's also there's also let me draw a little dotted line here there's that angle but then there's also this angle there's also this angle which is going even 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 well of course we want the one in the spring but if we just blindly apply the inverse cosine of negative zero point two that's going to give us this one that's going to give us this one right over here we can verify that so let's see inverse cosine of negative zero point two is one point seven seven remember this is zero this is PI right over here so this is 3.14159 and this is one point seven seven 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 one point seven seven radians 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 two pi and then subtract subtract one point seven seven so we could say two pi minus one point seven seven roughly to get this angle so let's do that so let's take let's take two two 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 four point five one one and we can we can kind of make sure that that's the right thing because that's going to be between PI and two pi two pi is 2 times 3.14159 so it's going to be six point two eight 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 day's going to be well 2 PI over 365 days is going to be equal to this number four point five one one so let me write that down so this right over here is going to be approximately equal to four point five one one so we scroll down a little bit we can say 2 PI over 365 times the days after January seventh is approximately going to be equal to four point four point five one one to solve for days we can just multiply both sides times the reciprocal of the coefficient so we're going to multiply both sides times 365 over two pi that's going to cancel that's going to cancel and now we can use our calculator for this so that's four point that's actually we got much better precision there so let's take our previous answer times 365 we deserve a drum roll here divided by two divided by two pi and we get 200 and if we round to the nearest day 262 days after January 7th so 262 days and we are done