Hey friends, I hope you're doing very well. First video we're going to see how to solve a trigonometric equation. But first of all, I invite you not to watch this video, but to first watch the previous video in which I made an introduction to trigonometric equations. Well, because the idea is for you to understand everything. No, I mean, no, not to learn the steps by heart, but to understand. And I explained this in the previous video, what is a trigonometric equation? What is solving it? How is it solved? And the important topics that we need to solve it. For example, t talks about functions of notable angles, and also the trigonometric circle, we must understand it well so that this seems easy to us. Ready, if you've already seen that video? Well, I invite you to solve this exercise as practice because it's the first video. Well, we're going to solve the simplest exercises in which there will be only one trigonometric function, that is, you will find only once the sine of X or the cosine or the tangent or the cotangent or the secant or the cosecant Yes in these cases where there is only one trigonometric function For example here that only says sine of X once and everything else are numbers, then the only thing we do Is clear that sine of X So what do we do here we observe that there are two terms one that is 2 multiplied by the sine of x and the other term that is the TR So that term we pass to the other side in this case since it is adding it becomes subtracting No here we would have on the left we would have 2 multiplied by the sine of X equal and on the other side it would be 4 but the 3 becomes subtracting then 4 - 3 gives 1 I know that I can skip a step we do not continue clearing the sine of X then this two that is multiplying goes to the other side to divide Then we would have that the sine of X is equal to 1/2 Yes surely if you have already seen the previous video this seemed simple to you something important If you had a calculator we simply apply the arcsine and we already found an answer but for the other answers Well we have to see well I I recommend that you see the trigonometric circle so that it seems easy to you ready In these cases in which we do not know sorry in which we do not have a calculator how do we want to Find the angle yes of which the sine of that angle is equal to 1 half, well we have to resort to our table of notable angles which is this yes Then we already saw that our function says that the sine of something must be 1 half then in the table we look in the sine column What is the result in which it says 1/2 here it says 1/2 that is, the angle of which the sine is 1/2 is the angle of 30 gr Yes, that is, the sine of 30 gr is 1/2 Remember that the answer is also generally given in radians Remember that 30 gr is the same as pi6 no Then we already found the answer to our equation The answer is 1/2 but since here we write it no then x would be equal to I said that 1 half was equal to 30 gr not that Remember that it is also equal to pi seos radian or 1/6 of pi radian any of both you can say them but Generally Well in trigonometric equations Remember that there are infinite amount of solutions and well if they don't tell you anything but simply to solve the equation Generally one gives the only one the first answer Not the one given in the trigonometric table and that's it But sometimes they will tell you to give the results between 0 and 180 gr that is between when x is worth between 0 and pi or sometimes they will tell you between 0 and 360 gr or sometimes Sometimes they will tell you to leave all the answers indicated ready I will teach you How to find the answers between 0 and 360 gr that is between 0 and 2 pi that is when the exercise tells you 0 less than or equal to x less than or equal to 360 gr If I told you 180 gr well it is much easier no because well it is less in this case we are going to find the other answer for that what do we do we already know that a solution is 30 gr so in our trigonometric circle we are going to draw the angle of 30 gr that there is no need to that is so exact no, we already saw this in the previous video no, I already clarified it with everything with all the details well almost all the details in our trigonometric circle we draw in normal position the angle of 30 gr so I already know more or less where it is then here well if you want you would take the protractor more or less you measure 10 20 and 30 gr it is more or less here Yes there is the angle of 30 gr that is, this angle measures 30 gr yes Remember that in the equation the solution was the solution of the sine of the angle was positive So let's look for which In which quadrants the sine is positive then in our trigonometric circle that you should already know look that in the first quadrant the sine is positive but we look in which In which other quadrant the sine is positive to translate this angle of 30 gr also in the second quadrant also the sine is positive in the third the sine is negative so in the third quadrant the answer is no good to me because here it would give a negative sine in the fourth quadrant Nor because the sine is negative So since these are the only two quadrants in which the sine is positive, then we also draw this angle here in the other quadrant, that is, here in this second one, in the second quadrant, obviously drawing from here, not starting in this quadrant, starting from the x-axis, from the x-axis, we draw the angle of 30° more or less, it would be like 10 gr, 20 gr, and 30° more or less around here. Yes, something like that would be the angle of 30 gr, that is, we measure the angle here in that quadrant starting from the x-axis, not upwards. If it were in this quadrant, we would measure it 30 gr downwards. Or if it were in this quadrant, we would measure it downwards. Yes, now to give the second solution, look, the first solution was 30 gr. Remember that we always have to give the solutions in the angle in normal position. This angle in normal position is already 30 gr, but this angle in normal position is not 30 gr because, remember, when it is in normal position, we would be talking about it being measured from the positive semi-axis of the x's until we get here. this is the angle that matters to us not the one we drew but the one that is in the normal position that reaches the same final side do not remember that well this angle what would it be Well look that from here if we make the complete turn Up to here it would be 180 gr So what would this other angle be Well it would be 180 gr we subtract the 30 that went up here or in other words the answer would be 180 - 30 that is 150 gr What does it mean that in our equation there is not only one answer that x is 30 that is one answer But there is another answer the other x that generally one writes x number 2 and x number 1 would be 150 gr that Remember that it can also be written in radians it is better to write it in radians then 150 gr when writing it in radians it would be 5/6 of pi radian yes something important Here we finish something that I like is that we check if yes these These are the solutions to our equation How is it checked in a very simple way using the calculator So I invite you to take your calculator and let's do the following operation look here it says the equation that we had to solve was 2 multiplied by the sine of something + 3 it had to give us 4 so what I like to do is do this operation and obviously if this operation with the answers that it gave me equals 4 like in this case well it means that these are the solutions to our equation so let's do it look what we are going to do 2 multiplied by the sine of the angle for example we are going to check if 30 is indeed a solution and we add three then I tell you again we are going to write 2 by the sine of the angle + 3 then we write in the calculator 2 sine of the angle which in this case the angle that we are going to check is 30 gr We close the parenthesis + 3 And that how much it has to give us it has to give us four and look that effectively it gave us four What does it mean that 30 gr was indeed one of the solutions to this equation now how do we verify the other angle we already have there written in the calculator 2 by the sine of 30 + 3 So now instead of 30 what do we have to write to see if the angle of 150 It also works So I like with the little arrow to go back I erase the three and write 150 s easy way to verify quickly we press equal and again it gave us four which means that the angle of 150 gr if it was a solution to our equation That is, we have already found the two solutions and well with this I finish my explanation but as always finally I am going to leave you an exercise or rather two exercises for you to practice if you have already seen the previous video this will seem simple to you if you have not seen it then I invite you to watch it I do not invite you to solve these two equations so that you can practice more things most likely you will suddenly have some doubt That is good because well you will solve the doubts when you see and compare what I am going to do ready So I invite you to pause the video solve these two equations and compare with the answer that I am going to show you in three two one in these two exercises the same thing happened as in the one I explained. Look, here there was only one trigonometric function cosine of X here there is also only one sine of X in this case we solve the first one to clear the cosine Well the two that is multiplying goes on to divide the cosine of X was already cleared So now we are going to look in our trigonometric in the row in Yes in the cosine row to see where it gives í de2 over 2 then we look in our table in the sorry column in the cosine row √2 over 2 what is the answer is 45 gr which Remember that 45 gr is the same as pi cu4 No I explain it with angles in degrees because well it is easier to understand no But remember that the answer is given in radians too because generally the no in the two no no then the cosine That is that the x is 45 gr so that is why here I wrote the x number 1 is 45 gr which is the same as pi cu4 what do we do Well here I explain it with the trigonometric circle but generally what I do after is I close my eyes I imagine the circle Ah I know that I have to subtract 180 from 180 I subtract the angle and now but for now so that we practice in our trigonometric circle we locate the angle of 45 gr so here I have it we locate the 45 but since there the sine was positive that obviously it had to be positive because in our equation here it says Ah sorry the cosine the cosine is positive Yes even in the answer I was wrong But well that helps us learn smarter here in the cosine It was ra2 over 2 that is, we look for the cosine by graphing here No here it was ra2 over 2 which was 45 gr So we locate the 45 gr don't look at this because this is wrong this is if it were the sine now the cosine that look here I marked the positive cosine was in the first quadrant but we look in another quadrant in which the cosine is also positive in this case this quadrant is of no use to me because the cosine is negative this quadrant is also of no use to me because the cosine is negative But what is useful to me is quadrant number four, that is, this angle of 45 gr we do not move it to the second quadrant but rather we move it to the fourth quadrant, that is, here in the fourth from the x down towards the fourth quadrant 45 gr That is, this is where I locate my 45 gr now what is the other answer look what I like is that you do not learn it by heart but I put it is logical how do we find this angle in normal position and I remove the other one because it does not interest me in normal position This angle is to start from here and turn it around like this no Remember that is the angle in normal position So what would we do in this case to the complete turn that is 360 we remove these 45 and what do we have left then here we would do 360 gr - 45 gr And that gives us 320 300 15 gr that would be the second answer 315 gr Remember that can be verified here I also wrote it in radians Remember that you can verify in this case in the equation What was it saying that 2 multiplied by the cosine of X was equal to i of 2 because some calculators give you the answer root of 2 but if not Well you would have to find out first How much is root of do is 1 point 40 and something I think Well it should give you root of do ready I already verified this Then it is correct let's go with the second in the second There is also only the sine Then we clear that sine the 5 that is adding goes to subtract we would have 7 - 5 which is 2 the four goes to divide here we can simplify half of 2 is 1 and half of 4 2 That is to say that here it says that the sine of theta which in this case the variable is theta is 1 half so we look in the column of the table and we observe where the sine was a half which we had already done was in the angle of 30 gr then in our trigonometric circle we locate the angle of 30 gr that in this case in the first quadrant the sine is effectively positive we look for In which other quadrant the sine is positive in this case it is in the second quadrant Now we move it to the second quadrant always starting from here we go up 30 gr remember that for the other angle Well the angle has to be in normal position that is to say it would have to be this angle not that Remember that it is 180 gr we subtract 30 then Well I don't do it anymore 180 - 30 that is 150 gr that you can write it as 5 pi sexes radian also the verification Well in this case we do all this and this operation well it has to equal 7 on the calculator and well I hope you liked my way of explaining and if so I invite you to watch the other videos of the course so that we can do exercises Now yes more difficult Here I also leave you Some videos that I am sure will be useful to you Do not forget to comment on what you want share this video with your classmates and they will surely appreciate it I invite you to subscribe to the Channel to give a good like to this video and not being more bye bye