trigonometry is a branch of mathematics concerned with the relationships between angles and triang sounds like a whole lot of yapping to me and I'm sure it is to you as well so I'm going to compress as much of the topic as I can in this video the entire not really explain 12 minutes there we go we begin with a concept which was probably the first one my smooth brain was ever shown and that is the right angle triangle it's a right angle because one of its angles is always 90° and that allows for a particular equation to hold true for any variation of this right angle a² + b² = C 2 for practicality let square root both sides length C is equal to theun of a s + b s what this thing means is that if we are given two lengths of a right angle we can find the missing third length the longest length is always the diagonal one called the hypotenuse and if we put an angle at one of the ends of the hypotenuse the length opposite to the angle is the opposite and the length adjacent to it is the adjacent that's crazy so we are given the lengths of the opposite and adjacent three and four and we wanted to find the hypotenuse C all we do is check this in the calculator and you get five why is it equal to five well because the square of three is 9 and the square of 4 is 16 9 + 16 = 25 and the Square < t of 25 is 5 well hang on a second if you've done any form of algebra you should know that the square root of a positive integer is always plus or minus the value but because we are dealing with geometry lengths are always positive so it's not Plus orus it's just five if you were given the hypotenuse in one of the other links simply take the original thing that we figured out and put the values that you know in it whether the hypotenuse is being subtracted by the other thing you know and then do the same calculations and you get your missing length so when I first heard Saka TOA I thought it was the name of a Greek mathematician like [Music] Pythagoras right yeah no contrary to unpopular belief it literally just means this s opposite hypotenuse cosine adjacent hypotenuse tangent opposite adjacent clearly each of those side legs have to do with a different angle functions and one of the easiest ways to remember the processes behind using these to work out angles of side length of right angles is by putting them in their own little triangles like these these are formula triangles now I know they're not necessarily the best way to learn math in the long term but for simpler Concepts like Sak TOA it's pretty useful so if you wanted to find the angle Theta you would first look at what length you know say you were given the opposite and the hypotenuse then to find this angle just take the inverse sign of the opposite divided by the hypotenuse and you follow the same process for finding the angle with the other legs and if you wanted to find either the opposite or adjacent depending on which other length you know just multiply that other length by your angle the unit circle s and cosine are waves and the reason for this is something called the the unit circle it's exactly how it sounds a circle with units and these units involve cosine sign L and radians what are radians are pretty much a type of angle that are related to the radius of a circle and for the actual definition of the radian that's circle is the unit circle so the unit circle has a radius of one this means that one radian forms an arc length in the circumference that is the same as the radius therefore if you increase the length of the Ark the amount of radians increases also so you could ask how many radians are in a 4 360 well keep lengthening The Arc Length until it is the same as the circumference and that gives a full 360° worth of radians the circumference of a circle is defined as 2 pi * R this means that there are 2 pi amounts of radiuses lay to create the full circumference and so doe 2 pi is the amount of radians in the full 360° of the the unit circle yeah okay pal but that doesn't explain what s and cosine have to do with it well the unit circle is broken up into four quadrants with increasing radians going anticlockwise in which these angles are conveniently broken up into rational fractions of Pi take 45° in the first quadrant for example now 45° is actually the same as pi over 4 radians we know this by multiplying pi over 4 by the conversion quantity 180 over pi and you get 45 what this means is that if you wanted to convert any degree into radians divide it by 180 over pi and if you wanted to do the conversion the other way just multiply it by 180 over Pi but let's imagine that you wanted to find the coordinates of the ends of an arc length now it's actually pretty simple as long as you know the radius of the circle and the radians that subtend the arc let's say the arc goes from 30° to 60° which is the same as piun / 6 to piun 3 to find these points here all we have to do is compute co sign of Pi / 6 to find this x coordinate and S of of Pi / 6 to find it Y and then do the same for Pi / 3 and Bam you have the coordinates as long as you multiply each one by the circle's radius otherwise that point will literally end up in the backrounds interestingly because of two types of right angles called the 30 60 90 triangle and the 45 4590 triangle these key fractions of Pi will return fractions of < tk2 < tk3 and one I don't want to keep saying radians and denote the angle so I'm just going to say angle Theta instead because COS of theta is the X and S of theta is the Y and the gradient of a line is defined as the difference in the Y between two points on the line divided by the difference in the X this means that we can also find the gradient of this line by simply dividing sin Theta by cos Theta I'm going leave this diagram here for a couple seconds it's important to notice that the coordinates of the reflected angles are obviously just s inversions of each other and because you guys seem to like my beat boxing in the last video I'm going to do a little more of that while you write this down okay I get it my big boxing sucks but do you know what doesn't suck brilliant.org the sponsor of today's video my favorite hobby was procrastination up until I downloaded brilliant on my phone which allows me to learn a massive range of content in maths science and computer science right from my pocket they've got bangers like exploring data visually calcul geometry thinking code literally the whole shebang through these you learn by doing building skills and problem solving analytical thinking and overall a daily learning habit that's much better than scrolling Instagram reals or even worse yet Reddit oh my God Reddit one of the best parts of brilliant is that the courses have the main content mixed along with relevant practice problems in order to promote Active Learning and I know I've said this in previous videos but brilliant is six times more effective than watching boring ler videos so that sounds especially based to try everything brilliant has to offer for a full free 30 days use my link brilliant.org slind why and you'll also get 20% off an annual premium subscription L run run run run unit circle is great and all but it's not a triangle it's a circle where the triangles come in well there's actually two very important laws to know for geometric problems and they are the law of signs and the law of cosiness remember this use the law of signs if you know two angles in one side or two sides in the angle opposite to one of them and use the law of cosines if you know two sides in the angle between them or the three side okay but that doesn't explain what they actually do if you were given this triangle here which law should you use to find length x the answer is cosine law because we know two side lengths in the angle between it traditionally this law takes the form c^ S = A + B S - 2 a * COS of capital c which is the angle but let's replace these variables with what we've been given because X is the length that we want to find we replace c^ S with x² a squ with 5 squ and b^ S with 4 squ now because the side length X the little C in the original equation The Big C is the angle opposite to length C which is 92° in our problem for these triangles we deal with degrees not radians because we want to find X we can square root both sides and our final equation for x becomes X = < TK of 5 + 4 - 2 * 5 * 4 * COS of 92° which is roughly equal to 8.13 Let's imagine a new triangle where you know two lengths A and B and the angle opposite to them and you want to define the angle opposite to B well we're going to have to use a sign rule which states that s of capital A which is the angle divid a equals s of capital B ID little B which is effectively saying that the ratio between an angle as an opposing length is equal for all of them within that triangle just like we did for the law of cosiness we can substitute an our values that we know and then solve for the angle B which in our case is x s of 48 over 5 = S of x / 4.5 now the solution from here is pretty simple we just want to isolate [Music] X so multiply both sides by 4.5 to eliminate this division then apply the arc sign function to both sides which is the inverse function of sign inverse functions simply eliminate the effects the function denoted as the function to the power of negative 1 for example the inverse of f of f ofx equals x because the inverse of f cancels out the F function so yeah sign is canceled just down to X and so x equals the AR sign of 4.5 * s of 48 over 5 which is roughly equal to 41.987012 better what values are substituted in the first one we're going to look at is the Pythagorean identity but to understand it let's first go back to the view the center of the circle is called the origin and it has the coordinate 0 0 if we choose a random point on the circumference and draw a line from the origin to that point it actually forms this right angle where the x coordinate of the point is the length of the base and the y coordinate is the length of the side because this is a right angle we can apply the Pythagorean theorem X2 + Y2 = 1 2 which is 1 now I'm hoping that you haven't just been playing fortnite this whole time you can actually remember from earlier that the x coordinate for the unit circle is defined as cos Theta and Y is sin Theta this means that the COS s theta plus sin s theta equals 1 and that right there is the Pythagorean identity and then you have two other pythagorean identities one for it divided by sin square of theta and the other by cos Square Theta there's also the reciprocal identity so 1 / sin Theta = cose of theta 1 / cos Theta = secant of theta and 1 / tan thals cent of theta which is also the same as cos Theta over sin Theta because tan Theta equal sin Theta over cos Theta there is something else called angle sum and difference identities which are exactly as they sound if you have sign of two angles added or subtracted from each other it is equal to the S of a * COS of B plus or minus COS of a * s of B for cosine that's practically the same but instead it is cosine of a * cosine of B and then the minus plus sign which is like the dependent version of plus or minus so if it's COS of a plus b then it's that minus that but if it's COS of a minus B it's that plus that now even more confusing yet tangent of a plus orus b equal tangent of a plus or- b okay that's pretty sure but then it's divided by 1 - plus of tan of a * tan of B all righty now we have double angle identity so if you had the sign of double angle Theta so s of 2 * thet that would be equal to 2 * sin Theta * cos Theta because you have COS of 2 * thet = cos Theta - sin Theta and finally tangent of 2 * thet = 2 * tangent / 1 tan squ of theta well we've covered quite a few things in trigonometry and I know it's not actually everything of trigonometry there's always parametric equation there's tons of stuff to cover but this will give you decent framework to work from to move on to a more high level content so they covered the Pythagorean theorem soak TOA the unit CC the unit circle LS of triangles trigonometric identities and of course brilliant.org which you should check out while using my link in the description I hope you guys enjoyed good luck with the trigonometry and I'll catch you in the next one bye this is terrible it's terrible it's bad it's not even good why do people watch this yeah okay how much guest do we have like not zero