in this video we are going to look at solving trig functions and this is a key concept in every single IV math exam so we need to be familiar with how to solve a trig function so a simple example of a trig function needs to be solved is if we have sine of X and this might have a bracket around the X or it could be a theta either or some variable and it's going to be called some fraction so sine of X is equal to 1/2 and if you are familiar with your trig ratios and there is another video on trig ratios we we looked at sine cos and tan of certain angles and they equal ratios that we need to remember and we need to remember them in non a calculator exams so 1 & 2 is part of one of the magic triangles and that magic triangles helps us solve these types of questions without a calculator so maybe revisit that video if you haven't seen these ratios before but what I'm going to do is if I have my my trig function in this form where sine or cos or tan of something equals a ratio I need to look at these two terms and think what triangle was that part of and this is part of the triangle where has one two root three where this is sixty degrees this was 30 degrees and 60 is the same as PI on three in radians and 30s the same as PI on six in radians okay sine is opposite over hypotenuse so what we want to do is want to think about sine of what angle you want to be looking at the angles in our triangle sine of what angle is going to be one over two and this is the challenging step we need to try and decide which angle it's going to be and hopefully you saw that PI on six would be our angle because the opposite to PI R 6 is 1 and the hypotenuse is 2 and sine is opposite over hypotenuse ok so once we add once we have identified which angle it is I like to call this my tool angle my tool angle it's at all X so you can say theta it's going to be X will be PI on six now I call it a tool angle because often our question has a given domain a domain where we need to solve our function for X and an example would be zero to X to PI and this means that we're going to have multiple answers within our given domain so with that tool angle I know that all of our angles will be PI on 6 it will have some sort of link to pile on six will be politics away from the x-axis and I'll show you what that means if we draw our unit circle with our quadrants all stations to central ASTC we need to think about okay sign gave us a positive ratio of 1 on to what are the positive sign quadrants positive because we had a positive ratio and the positive sign quadrants are the S quadrant and of course the a quadrant based and all s 10 for sine T is the 10 see it's because so once we know that our our trig function here sine was positive 1/2 we need to look at the positive sine quadrants if this was negative 1/2 we would be looking at the negative sign at quadrants 10 C so our answers within our domain which is one full circle this is 0 PI 2 pi 3 pi into 2 pi will be PI on 6 up in either of these two quadrants so if we start at 0 and up and if we go all the way over to PI and we go up these two red lines here this one here and this one here will be our two solutions for X and this angle is simply 0 plus PI + 6 which will be part 6 and this one's PI minus PI + 6 you might need to do some fraction work there PI minus PI + 6 make sure we we know how to do this just by doing a simple common denominator this would be 6 PI on six - pylon six which is five pi on six so they will be the two solutions for X for this question in our given domain so therefore X will be equal to PI on six and five PI on six now there aren't any solutions in the third and fourth quadrant because there will be negative sign values and hopefully through the trig ratios for video in the unit circle video we remember that sine is the Y value of any angle and if you have a look at the two red lines that I've drawn the Y value is the same it's going to be positive 1/2 compared to its hypotenuse so these are the two solutions I am just going to quickly draw what the graph looks like and how we can picture this if it was a graph so a sine of X function just simply looks like this where this would be PI this would be 2 pi this here would be PI onto this here would be 3 PI into and what this question is saying graphically is when is this function equal to positive 1/2 positive 1/2 would be here because we know sine goes up to 1 and it's going to be equal to 1/2 2 times it's going to be equal to 1/2 here and here and if we go down and find those two values this will be PI on 6 and this will be PI minus PI + 6 which is 5 point 6 so we can either solve it using our magic triangles and out in a circle or graphically this is the times that it equals 1/2 in the first period okay so that's an example of solving a trig function your questions probably won't be as simple as just sine of X is equal to 1/2 they might try and give you a little bit of a tricky equation to begin with so an example might be 2 cos x and then plus 3 is equal to 2 and I'll say solve for X and you might think well that's a lot different to what the example up here was but all they want you to do is try and rearrange this equation to get some trig ratio so in this example we would put three over two cars of X B we'd subtract 3 from to be negative 1 cos of X would therefore be negative 1/2 and then we would just take we would look and bassy which angle where cos is a positive 1/2 which would be 60 degrees because it Jackson over hypotenuse then we consider the negative we'd look at the two negative cos quadrants which would be this one and this one and we do the same process as we did here so we we can just rearrange our equation to get our trig function equaling our ratio okay I suggest you practice a few of these questions so good luck