This is a video about inverse trig functions, part one. I'll start off by talking about the graph of y equals sine x. This is the graph of the sine function. This is two periods of the graph. It continues going on in the rightward direction forever and the left hand direction forever. The domain of the sine function is negative infinity to infinity and the range is negative one to one. The issue here is for a function that have an inverse, it must be one to one, meaning it must pass the horizontal line test. If I was to go upward on this graph with the horizontal line, I would touch multiple points at the same time, which means that the function cannot have an inverse. So what I have to do is I have to restrict the domain to the following, negative pi over two to pi over two. So I'm gonna cut the graph from negative pi over two to pi over two and this leaves this upward diagonal looking figure right here. That's the only thing left of the graph. So negative pi over two to pi over two is my domain for this function that have an inverse. This is what makes the function one to one. So as a result, the inverse sine function, sine inverse of X or inverse sine of X. The sine function, when the domain is restricted to negative pi over two to pi over two and the range is negative one to one, it always is. The inverse sine function, its domain then becomes negative one to one and its range then becomes negative pi over two to pi over two. The roles of the domain and the range switch places. The range of the original becomes the domain of the inverse and then the domain of the original becomes the range of the inverse. The reason why is because trig functions as we have come to learn are applied to angles and the answer is a measurement. So I plug angles into the sine function and out comes some measurement from negative one to one. Inverse trig functions, they're applied to measurements. So in the case of the inverse sine function, they're applied to negative one to one and then the result is an angle. So we're working backwards now. We're gonna be given a measurement and our goal is to find the angle which gave that measurement. That's what we're going to be doing. So let's talk exclusively about the inverse sine function. Find the value of each of the following. So inverse sine of one, what this means is, sine of what angle theta is equal to one. So where does sine at what angle the sine equal one? I'll write this out in words for this example or this part here. So I'll say where is the sine function equal to one. Where does the sine function equal to one? Now let's think about this for just a minute. I'm gonna draw my unit circle. I'm gonna focus on my key points for this part here. So remember I have one zero upper portion here. I have zero one left hand portion. I have negative one zero bottom portion. I have zero negative one. Where is sine equal to one? So remember my angle that is spit out from inverse sine has to be between negative pi over two and pi over two. That would be negative quadrant four, the quadrant one. So somewhere in this right hand side of my coordinate plane. Now remember sine is the second coordinate. Which of these ordered pairs has a second coordinate of one? And that would be pi over two. So in this example, theta is pi over two. So inverse sine of one is equal to pi over two. The original angle was 90 degrees. All right, moving on to our next example. Inverse sine of negative one half. Now remember this value that we get from this inverse sine function has to be in either quadrant one or quadrant four. So I have a choice. It's going to be quadrant four or quadrant one. However, since sine is negative, we are in which quadrant it all trig functions are positive in one, sine is negative and quadrant four. Only cosine and secant are positive. All right, so we are in four. We are in quadrant four. Keep that in mind as I go through and I say, okay, sine of what angle is equal to negative one half? So I'm going to draw a triangle here. In this triangle, I'm going to have some sort of reference angle here. And that angle will be moved around to the appropriate quadrant. So sine of theta equals negative one half. Sine is opposite over hypotenuse. Opposite's one hypotenuse is two. So one, two, which means the third side has to be square root of three. So this is a 30, 60, 90 triangle. What angle is always across from the side with measure one? Well, the angle is pi over six or 30 degrees. But we need to move this to quadrant four. We must move to quadrant four. It has to be negative quadrant four because remember we have to be between negative pi over two and pi over two. Negative pi over two to zero is quadrant four, the negative part of it. And zero to pi over two is quadrant one. So to move an angle to quadrant four from quadrant one, you can simply just make the angle negative. So if an angle is positive in quadrant one, you can move it to quadrant four just by making it negative. So my angle is actually going to be negative pi over six. Theta is negative pi over six. So the answer to this inverse trick expression is negative pi over six. That is our answer. Next, part C, inverse sine of square root of three over two. Now my issue is when I go to draw my triangle, do I know anything that has a square root of three and a two? And the answer is yes, that would be 30, 60, 90 triangle relationships here. So we'll figure out what the answer is, but remember I have to have a final angle that rests in quadrant one or quadrant four. So I have a positive value here since sine is, since sine is positive, we are in one. We are in quadrant one in this case. So I have sine of what angle, that will be a quadrant one angle is equal to square root of three over two. Let's draw our triangle. We have some mystery angle we're trying to find and sine, it's square root of three over two. Opposite is square root of three, hypotenuse is two, which means the other side would have to be one. Who is always across from square root of three? What is my mystery angle? The angle is, well, it's a 60 degree angle. It's pi over three. And this is in quadrant one already. So it can be my answer because I need angle in quadrant one. So theta is equal to pi over three. So I have pi over three or 60 degrees is my answer. So this is using the inverse sine function to find original angle measures. Now we're gonna go in and we're gonna talk about all of the other inverse trig functions. But first I want us to kind of have a throwback and a review here of how to find reference angles. For any angle theta, the reference angle gamma is found from the following. Quadrant two.  Quadrant three. The reference angle gamma is equal to theta minus 180. Quadrant four, the reference angle is equal to 360 minus theta. Therefore, I can do the same thing. If I have an angle that is in quadrant one, I can move it back to its original quadrant using these formulas because that's what I'm doing here is I'm getting quadrant one angles and then I have to either remove them to quadrant four, I'll have to move them possibly to quadrant two here in a second as you'll see. So for instance, if I look at the quadrant two reference angle formula, and this is the one we will be using the most, I have gamma is equal to 180 minus theta. You add theta to both sides, you get theta plus gamma is equal to 180. Then you take away gamma from both sides and then you'll get theta equals 180 minus gamma. And that is exactly how we get this first formula in the bottom part of the slide here. Quadrant two, the original angle is equal to 180 minus the reference angle or pi minus the reference angle. All of the formulas on the bottom half here can be found by rearranging the formulas on the top half. So what this shows us are the inverse trig functions of the original six trig functions we originally studied. As you can see, sine function negative one to one, range is negative pi over two to pi over two. So this would be quadrant one and negative quadrant four. Cosine function that domains negative one to one but the range is zero to pi now. So we take the cosine function, we cut it from zero to pi. That's what makes it one to one. That's why the range of the inverse cosine function is zero to pi. So that would be quadrant one and quadrant two. Inverse tan, the domains negative infinity to infinity, the range is negative pi over two to pi over two. So this would be quadrant one and then you have negative quadrant four. Inverse cosecant, once again, you have quadrant one. You have negative quadrant four. That's where your answers can be. So the only places. And then inverse secant, that would be quadrant one and two. And then there's inverse cotangent, which the answer must also be in quadrant one and quadrant two. So you have two options here. You have three trig functions which their inverse gives you angles in quadrants one and two. Then you have three inverse trig functions which give you angles in quadrants one and quadrant four, the negative quadrant four. So let's work some examples. My first example is gonna be inverse tan of square root of three over three. My issue is this. I do not have a right triangle relationship that uses square root of three and three. The reason why this looks unfamiliar is because somebody took the original measurement that was in here and rationalized it. So what this is is saying tan of what angle is square root of three over three. I wanna undo the rationalizing here. So multiply by square root of three over itself. This gives me tan of theta is equal to three over three square root of three, which is one over square root of three. This looks more familiar. I know a relationship that has one and square root of three and it's a 30, 60, 90 triangle relationship. So let me go ahead and make my triangle with my mystery angle here. Tan of this angle is opposite as one adjacent to square root of three, which means hypotenuse has to be two. So looking here, who's always across from one? Well, that would be 30 degrees. That would be pi over six is our angle. Now the question is inverse tan, our answers have to either be in quadrant one or negative quadrant four. How do I figure this out? Well, look at the sign of the original value. We had square root of three over three. First, to reiterate, your final answer must be in the interval negative pi over two to pi over two. It's quadrant one or negative quadrant four. Now, question is which one is it? Since tan of theta is positive, since it's positive, theta must be in quadrant one. So whatever our mystery angle is, this question mark is just saying the reference angle. The actual angle theta, the answer to this inverse trig expression is going to be quadrant one. So since pi over six is already in one, so I'll write already in one, we don't have to do any moving around. Our answer, inverse tan of square root of three over three is pi over six. Next, inverse cosine, I have negative square root of two over two. When we're talking about cosine, inverse cosine, our final answer must be in the interval zero to pi. Zero to pi. It has to be in the interval zero to pi. That's quadrant one or quadrant two. I have cosine theta equals negative square root of two over two. Since cosine is negative, so since cosine is negative, our angle, our answer, theta must be in quadrant two. Must be in quadrant two. Now, the issue is I have the square root of two over two and I don't know any triangles with the relationship that have square root of two and two in them. Once again, we're gonna have to go through and undo the rationalizing. Multiply by square root of two over itself. That gives me two over two square roots of two. That gives me one over square root of two. So now I have cosine theta equals negative one over square root of two, which this enables me now to make my triangle. So I make my triangle. I have my mystery angle, my quadrant one reference angle, I guess is what you could call it. Dealing with cosine adjacent over hypotenuse. Adjacent is one, hypotenuse is square root of two, which means this other side has to be one. This is a 45, 45, 90 triangle. So our angle, our question mark here is pi over four. But I have to move this angle to quadrant two. That's where my angle needs to be. So my original angle theta, using the formula from the previous slides is pi minus my reference angle, pi minus pi over four. This would be four pi over four minus pi over four, which is three pi over four. So the answer here, our theta, our angle, the inverse cosine function's value for this example would be three pi over four. That is a quadrant two angle. Inverse tan of zero. Anytime you see zero or in here, you should think about the key points. That's what we wanna think about here. I'm gonna turn this into tan of what angle is zero over one, zero over one. We need to have a ratio, so that way I can know what cosine is and sin is. My angle, my final answer, final answer must be in the interval negative pi over two, two pi over two. So now I'm gonna draw my unit circle. Looks a little bit nicer than the first one we did on this lecture, label these key points and which of these points has sin of zero and cosine of one. Remember cosine is the first of the ordered pairs and sin is the second. So one zero is what I'm looking for. And that puts me right here at zero. So my angle here, theta is zero. So that is how to evaluate inverse trig expressions.


Coming up on this edition of Inside Look. It has been burning for a week now, but the massive Thomas fire shows no signs of letting up. That has now become the fifth largest wildfire in California history. We're almost to 2018, but the 2017 fire season is still in full blaze. Southern California now feeling the heat. Some scientists are saying Southern California is literally burning up. We'll take you to the fire lines and show you how California's mutual aid system is once again coming to the rescue. Without the support of some of the states that are in here helping us, I don't know where we'd be. It was an alert that warned more than 12 million people of unprecedented fire danger in Southern California. We'll hear firsthand what prompted state officials to hit the send button. And we'll take a trip back to Northern California to get an update on the recovery progress from the October wildfires. I'm Brian May in the Cal OES newsroom. It has been a busy week and a tragic week as well. On Thursday, a 32 year old Cal Fire firefighter lost his life while on the line at the Thomas fire burning near Ventura and Santa Barbara. That fire is the number one concern for emergency managers and going into the weekend red flag warnings are saturating much of the state from the North Bay to the Eastern Sierras and Southern California. The Thomas fire, now the fourth largest fire in California history. It is more than seven and a half times the size of San Francisco larger than New York City and Boston combined. Our Cal OES information team is all over the California December wildfires. Jonathan Goodell is in Santa Barbara to talk with us about mutual aid. Cal OES deputy director Kelly Houston is in the state operation center to talk with us about the wireless emergency alert system. But we begin with Sean Boyd who is in Ventura County. Sean. Yes, I'm here to venture a County EOC where some of the managers here have received a copy of the alert that went out to select residents in the Santa Barbara area. This is what they heard. Residents only. And that's just part of the message. It goes on. But while those select residents are being allowed back into their homes, the firefighters are wondering what's gonna happen to the calm winds that they've enjoyed recently. Those are expected to pick up dramatically over the next couple of days. The Thomas fire in Santa Barbara and much of Ventura County is relentless. At more than 238,000 acres, it's on its way to the dubious distinction of becoming the largest fire in California's long wildfire history. But firefighters are relentless too, having tirelessly worked to contain 30% of the Thomas. After a quote, pretty good day Tuesday with minimal fire movement, strike teams monitor the fire in neighborhoods above the Highway 192 corridor. And again, now at night, just like the previous night, the fire activities picking up and we're doing some structure defense. Assistant Chief Mike Taylor knows the night has just begun. Around 3 a.m., he expects winds to shift down the canyon and threaten homes. He's just looking for the signs. What is it that you're looking for? The smoke's our biggest indicator. Obviously, we can feel the breeze but sometimes when you're tucked into these canyons, you know, we can have wind here that's just around the corner, completely different. It poses a lot of challenges. Experience tells him it's gonna be a long night. Daybreak reveals fresh burn scars and flare-ups, big ones, that could add to the 930 homes already destroyed. Helicopters ferry Bambi buckets to and from canyons, dumping water on fires that continue to burn. More than 8,000 mutual aid firefighters are in on the battle and more than 94,000 people have been affected or evacuated. The shelter here at UC Santa Barbara opened for many of them on December 6th. And it's amazing when it all moves together like a big puzzle piece, it makes a big, beautiful recovery operation and that's what we do. Stephen Klopp says the evacuees here are emotionally drained. Right now we have about 70 people last night we did it this year. The rec center is now home for Anfield and Michael Acosta. They were camp hosts at Los Padres National Forest until they were evacuated. I know there's people in the shelter right now who are in a bigger mess than I am, but for us and him being sick, it's really harsh. A few cuts away Norma Sears and her 92-year-old mother are relieved they don't have to spend nights in their car or inhale the fine particulate matter and ash that's inescapable outside. And the Red Cross will come by and just ask her how she's doing. And the added attention Norma's mom gets from Red Cross volunteers is a blessing, she says. Red Cross, I didn't know that much about it. These people are phenomenal. As night falls once again, they wonder how their town of Carpinteria will fare. After all, the Thomas fire has been 60 years in the making. It's been that long since a fire burned in Santa Barbara's front country. That type of fuel. It's also Amber Anderson's hometown. So what we have now is a lot of very old growth, dense chaparral, chaparral higher than 20 feet in some places, which makes it incredibly difficult for our crews to get in there and make any sort of hold in front of this fire and to do it safely. Well, increasing winds means increased air debris, particulates, ash, that kind of thing. But we've seen just about everybody wearing these N95 masks, that's a good thing. But what's not a good thing? Expected wind gusts over the weekend, up to 50 miles per hour. That's gonna pose some serious challenges for firefighters. Brian, back to you. All right, Sean, thank you. Fighting the half dozen fires in Southern California required a massive amount of resources and personnel. At the peak, Cal OES coordinating mutual aid of more than 2,700 firefighters, 1,200 of those coming from 11 different states. We continue our coverage now with Jonathan Goodell, who joins us from Santa Barbara. John? Thanks, Brian. Since October, more than 20,000 firefighters, some from across the Western states, have joined the fight against these wildfires, first in Northern California and now here in Southern California. They came here from remote towns in Alaska, from the Northwest and the mountain range, all here for one fight. The mutual aid system is designed on assisting others. It's built on neighbor-helping neighbor. California offered assistance earlier this year for fires in Oregon and hurricanes across the US. Now, other states are pitching in to help California. In Utah, we have a couple hundred firefighters covering the entire state. It's a massive state, just like California is. So we really rely on other folks from other states to come over and help us when we need it. Kinda, and the same idea is that when you guys need it, we'll be there for you too. Six states deployed resources through the Emergency Management Assistance Compact, while another was arranged via Interstate Compact. In all, 14 states have sent resources. And without the support of some of the states that are in here helping us, I don't know where we'd be. Introduced nearly 70 years ago, the mutual aid plan has grown exponentially. Resources are quick to respond, not just throughout California, but state to state, by ground and by air.  I don't think anybody intended that mutual aid would be in Siskiah County helping San Diego or vice versa, or to the extent that we're doing it so constantly. Kaiser Fire District in Southwest Oregon has spent a majority of the past few months in California. After crews responded to the October fires, Matt Dryden, Aaron Pittis, and their crew are now here in Santa Barbara to fight the Thomas Fire. We're here to help, so whatever everyone needs from us, we're here to do what we need to do. Happy to be here and ready to help. Normally, the fires we get are in one kind of little geographical area. This is spread all over the place where you drive five minutes from here and you're seeing burned places already and then we drive for a half hour to get to where we're assigned. We're still seeing the same burned places and just the magnitude is unbelievable. With containment of nearby fires, some fire personnel are transitioning to Santa Barbara. Shane Ward and his unit from Utah arrived overnight after battling the Lylac Fire Fire in San Diego. By the morning, his crew was working the Thomas Fire. We want to get there quick. We want to be able to help out as much as we can because it's no one wants to see home. We all have a lot of empathy as being firefighters and first responders. We never like seeing that, especially if we can do anything to help. That's just where our mindset's at. Let's get there, let's help these folks out. To put it into perspective, just how many firefighters have been out here in Northern California and now here in Southern California, 20,000 represents nearly a quarter of the population in Santa Barbara alone? Brian? All right, John, thank you. The red flag warnings and weather conditions in Southern California last week led leaders here at the Office of Emergency Services to take extraordinary measures. This wireless emergency alert, or WIA, sent to over 12 million cell phones in seven counties from San Bernardino all the way down to San Diego County. Kelly Houston, his deputy director here at Cal OES, he was involved in sending this alert out last week. Kelly, joining us from the State Operations Center. And Kelly, I know that alerts are sent out all the time, so what makes this alert that you guys sent out so unusual? Well, if there was ever a time, Brian, for the state to send out an alert to millions of people, it was this time, because the fire danger in Southern California was extraordinary, where there's seven counties down there that all were in red flag conditions. In fact, beyond red flag, it was worse than it has ever been. And we wanted to make sure that as people were going into the evening hours, that they were getting an alert and being aware of what the conditions were. You got a lot of attention from this alert to the point that the New York Times did an article on it, and the comments that followed that New York Times story on social media were almost all positive. What are you guys hearing or action-wise from people who got those alerts? Well, there was a lot of careful consideration into whether or not we were going to send out an alert to 12 million people, because sometimes an alert can be annoying to folks, or they feel that they may have been over-alerted. But we thought that with the extraordinary danger that was present in this case, and the fact that we just needed people to know that it was very dangerous conditions overnight, the reaction by folks was actually a bit surprising. We had a lot of support from people saying that they appreciated that we took the time to alert them, even if they weren't in the direct fire zone or somewhere where a fire was actually burning. Kelly Houston, Deputy Director here at Cal OES. Thank you for joining us. To learn more about wireless emergency alerts, you can go to our homepage, Cal OES.ca.gov. And before we go, we want to give you an update on the cleanup efforts that continue from the October wildfires in Northern California to date over 5,200 rite of entry forms have been processed over the seven counties that were affected by those October fires. Over 438,000 tons of debris have already been removed from the burn zones. That is equal to the weight of more than four full-size aircraft carriers. For more information on the October and December wildfires, you can go to wildfirerecovery.org. For all of us here at Cal OES, I'm Brian May. Thanks for watching. Visit our online newsroom at oesnews.com to learn more about this program and get the latest news and information from our team. Don't miss our next video on your Facebook timeline. Like our page and you'll get the latest posts as they happen. If you're an Instagram user, you can see the latest snapshots by following our Cal OES Instagram account. And Twitter users can get instant access to our tweets from across the state by following Cal OES.


Hello, everyone. Welcome back another AI video. In this one, we're in chat GPT, and I'm going to show you the new custom instructions. It's brand new. It just popped today, I believe. And here we go. So right now I'm in GPT 35. I'm going to switch to GPT 0.4 or GPT 4, not 0.4, just GPT 4 or GPT plus. And then I'm going to click on these three dots. And I'm going to go down here. And you're going to see custom instructions. I'm going to click on this. And then check this out. You got to make sure that this is turned on, enable for new chats. And then you get two options here. What would you like chat GPT to know about you to provide better responses? I don't know, I live in Canada. I have a dog named Kingston, whatever. I'm just winging it here just for funsies. And then how would you like chat GPT to respond? I would like to be spoken to like a 9th century English king. Whatever. I'm having some fun with it. Click save as I'm going to do here. And then presto. The next time, let's just test this out. What is my dog's name? Let's see what even happens here. Yes, the loyal canine companion hath been bestowed the honorable name of Kingston, Sire. Yeah, this is awesome. This is how you do it. Way to go chat GPT. You just made my day. Thanks for watching.


T. F. N. N. headline news update. Good afternoon everybody I'm Tommy O'Brien coming to you live from T. F. N. N. 12 noon Eastern time on Friday middle of the trading day and we got markets right around flat territory off of the highs S. and P's negative by one point right now trading at thirty two seventy four we were as high as thirty two eighty seven pre markets three about thirteen points off the high Dow Jones negative by eighty seven points at twenty eight thousand eight seventy two Nasdaq actually positive by five trading at ninety two oh eight we'll start things off and we're going to start it with a chart of the jobs jobs number coming in this morning non-farm payrolls at eight thirty a.m. that number one hundred and forty five thousand jobs added from the month of December just under what they were looking for which was about a hundred and sixty thousand you can see the breakdown we got a C. M. B. C. article up here the biggest sector retail underneath that leisure and hospitality education and health construction coming in at twenty thousand and from there the negatives mining minus nine thousand transportation minus ten manufacturing minus twelve the headline there though one hundred and forty five thousand jobs and the unemployment rate three point five percent jumping over the charts will start it off with the Dow the Dow the laggard of the major indices Dow currently minus eighty six points looking at the futures market twenty eight thousand eight thirty two we were actually above twenty nine thousand overnight Nasdaq one hundred currently trading at nine thousand and twelve we're about forty points off the high we had earlier in the session there's your S. and P. five hundred thirty two seventy three we're as high as about thirty two eighty seven overnight crude oil with some volatility today as well we have crude trading at fifty nine dollars and thirty one cents right now gold had some volatility on the jobs number this morning we got an initial spike higher gold pullback to under fifteen fifty now we've seen acceleration since about nine a.m. gold is traded from fifteen forty eight now trading at fifteen sixty one and the Euro U. S. dollar trading just under one twelve at one eleven nineteen terms of what else you have happening in the market the news coming out just recently Pelosi Nancy Pelosi the speaker of the house will be sending the articles of impeachment to the Senate next week that looks to be the case Boeing more emails emerging they do not shine a positive light on Boeing and jumping over to see how those shares are trading this morning this afternoon I should say as it is twelve noon Boeing down about one point four percent some of the other movies out there about L. Brandt's catching a bid again up about three percent and restoration hardware pulling back a bit. Stay tuned folks we got a replay this hour but we have Steve Rhodes live at one day white live at two Tom O'Brien live at three we'll be right back.