have you ever heard of the expression SOA what do you think this expression means in this lesson we're going to focus on right triangle trigonometry let's say if this is the angle Theta now there's three sides of this triangle that you need to be familiar with opposite to Theta this is the opposite side and next to the angle Theta is the adjacent side and across the box or the right angle of the triangle which is the hypotenuse that's the longer side of the triangle now if you recall this is a b and c the Pythagorean theorem applies to right triangles a 2 + b 2 is equal to c^2 we're not going to focus on that too much but just be familiar with that equation now let's talk about the six trig functions in terms of s cosine tangent opposite adjacent hypotenuse sin Theta according to soaa S is for s o is for opposite H is for hypotenuse sin Theta is equal to the opposite side / the hypotenuse cosine Theta is equal to the adjacent side/ the hypotenuse C is for cosine is adjacent over hypotenuse and tangent Theta TOA is equal to the opposite side divid the adjacent side so that's the tangent ratio it's opposite over adjacent now we know that cosecant is 1/ s so cosecant is basically hypotenuse divid the opposite side you just need to flip this particular fraction secant is the reciprocal of cosine so secant is going to be hypotenuse ID the adjacent side coent is the reciprocal of tangent so if tangent is opposite over adjacent coent is adjacent divid by the opposite side now let's say if we're given a right triangle and we have the value of two sides let's say this is three and this is four and here is the angle Theta find a missing side of this right triangle and then find the values of all six trigonometric functions sign cosine tangent secant cosecant Cent now to find a missing side we need to use the Pythagorean theorem a 2 + B2 is equal to c^2 so a is 3 B is 4 and we got to find Miss inside C which is the hypotenuse 3^2 is 9 4 4 2 is 16 9 + 16 is 25 and if you take the square root of both sides you can see that the hypotenuse is five now turns out that there are some special numbers there's the 3 4 5 right triangle the 5 12 13 right triangle the 8 15 17 right triangle and the 7 24 25 right triangle and any whole number ratios or multiples of these numbers will also work for example if we multiply this by two we'll get 6 8 10 that can also work or if you multiply by three you get the 9 12 15 triangle if you multiply this one by two you get the 10 24 26 triangle those are also special triplets they work with uh any right triangle now some other numbers that are less common but you might see are the 9 40 41 triangle and the 11 6061 so if you see some of these numbers you can find them Miss inside quickly if you know them so now let's finish this problem so what is the value of sin Theta so according to SOA TOA we know that sin Theta is equal equal to the opposite side divid the adjacent side and the part so s opposite to Theta is four and hypotenuse is five so therefore sin Theta is going to be 4 / 5 now cosine Theta is equal to the adjacent side / the iuse so we said four is the opposite side five is the hypotenuse and three is the adjacent side so in this case it's going to be 3 / 5 so that's the value of cosine now let's find the value of tangent tangent Theta according to TOA is equal to the opposite side divided by the adjacent side so that's going to be 4 / 3 so that's the value of tangent now once we have these three we can easily find the other three to find cosecant it's 1/ sin so just flip this fraction it's going to be 5 over 4 and secant is the reciprocal of cosine so flip this fraction secant is going to be 5 over3 coent is a reciprocal of tangent so if Coan I mean if tangent is 4 3 coent is going to be 3 4 and that's how you could find the value of the six trigonometric functions let's try another problem so let's say this is Theta again and this side is eight and this side is 17 find the Miss ins side and then use the completed triangle to find the value of the six trigonometric functions so go ahead and pause the video and work on this problem so first we need to know that this is the 8 15 17 triangle if you ever forget you can fall back to this equation so a is 8 we're looking for the missing side B and the hypotenuse is 17 8 s is 16 4 and 17^ 2 is 289 289 - 64 is 225 and we need to take the square root of both sides and the square root of 225 is 15 which gives us the missing side of the triangle so now go ahead and find the value of sin Theta cosine Theta tangent Theta and then cosecant Theta secant Theta and cotangent Theta so using soaa we know that s is equal to the opposite side divided by the hypotenuse so let's label all the three sides 17 is the hypotenuse 8 is the adjacent side and opposite to Theta is 15 so opposite over hypotenuse this is going to be 15 / 17 so that that is the value of sin Theta now cosine Theta is going to be equal to the adjacent side divided by the hypotenuse so the adjacent side is 8 the hypotenuse is 17 so cosine Theta is 8 over 17 tangent based on TOA is going to be opposite over adjacent so opposite is 15 adjacent is8 therefore tangent is going to be 15 / 8 now cosecant is the reciprocal of s so if sin Theta is 15 17 cosecant is going to be 17 over 15 secant is the reciprocal of cosine so if cosine is 8 over 17 secant is 17 8 you just got to flip it and cant is a reciprocal of tangent so cang is going to be 8 over 15 just flip this fraction and now we have the values of the six trigonometric functions and that's all you got to do so here's a different problem so let's say here's our right angle and this time this is Theta and let's say the hypotenuse is 25 and this side is 15 find the missing side and then go ahead and find the value of the six trigonometric functions so this is going to be similar to the three four five triangle notice that if we multiply everything by five we'll get two of the three numbers that we need 3 * 5 is 15 4 * 5 is 20 5 * 5 is 25 so we have the 15 and we have the 25 therefore the missing side must be 20 and you could use the Pythagorean theorem to confirm this if you want to so now let's go ahead and find the value of sin Theta so opposite to Theta is 20 the hypotenuse is always across the box it's the longer side so 27 is the hypotenuse and adjacent to 15 or right next to it is 15 I mean adjacent to Theta is 15 now sin Theta we know it's opposite divid hypotenuse so it's 20 over 25 which reduces to 4 over 5 if we divide both numbers by five 20 ID by 5 is 4 25id 5 is 5 cosine Theta is adjacent over hypotenuse so that's 15 ID 25 which reduces to 3 / 5 tangent Theta is opposite over adjacent so 20 over 15 which becomes if you divide by five that's going to be 4 over 3 now cosecant is the reciprocal of s so it's going to be 5 over 4 based on this value and if cosine is 3 over 5 then secant the reciprocal of cosine has to be 5 / 3 now if tangent is 4 over 3 coent has to be 3 / 4 and so that's it for this problem consider the right triangle in this right triangle find the missing side in this case find the value of x let's say the angle is 38° and this side is 42 so what trick function should you use in order to find the value of x should we use S cosine or tangent well relative to 38 we have the opposite side which is X and the adjacent side which is 42 so tangent we know it's opposite over adjacent so therefore tangent of the angle 38° is equal to the opposite side x / the adjacent side 42 so in order to get X by itself we need to multiply both sides by 42 so these will cancel so therefore X is equal to 42 tangent of 38 so we need to use the calculator to get this answer and make sure your calculator is in degree mode so tan 38 which is 7813 and let's multiply that by 42 so this will give you an x value of 32.8 now let's try another example feel free to pause the video to work on each of these problems by the way so let's say this angle is 54° and we're looking for the value of x and hypotenuse is 26 which trig function should we use S cosine or tangent so opposite to the right angle we know it's the hypotenuse and X is on the adjacent side relative to 54 so cosine is associated with adjacent and hypotenuse so therefore cosine of the angle 54 is equal to the adjacent side x / the hypotenuse of 26 so to get X by itself we got to multiply both sides by 26 so therefore X is equal to 26 cosine of 54° cosine 54 is .58 7785 if you multiply that by 26 this will give us the value of x which is 1528 here's another one that we could try let's say the angle is 32° and the hypotenuse is X and this is 12 12 so notice that 12 is opposite to 32 and we have the hypotenuse so this time we need to use the sign function s of the angle 32 is equal to the opposite side 12 / X so in this case what can we do to find the value of x what would you do what I would do is cross multiply so 1 * 12 is 12 and this is going to equal x * sin 32 next I recommend dividing both sides by sin 32 sin 32 / by itself is 1 so therefore X is equal to 12 over sin 32 12 / sin 32 is 2264 so that's the value of x in this particular problem now let's work on another problem so this time we need to find the angle Theta and we're given these two sides so five is opposite to the angle and four is adjacent to it so what trig function can relate Theta 4 and 5 we know tangent is opposite over adjacent so tangent of the angle Theta is equal to the opposite side which is 5/ by the adjacent side 4 so how can we find the value of the angle Theta if tangent Theta is 5 4 then Theta is going to be the inverse tangent or arc tangent of 5 4 and you simply have to type this in your calculator so type in Ran 54 and you should get an angle of 5134 de so that's how you could find a missing angle let's try another example feel free to pause the video and find a missing angle so in this case we have the adjacent side and we have the hypotenuse so therefore this is associated with cosine cosine thet is equal to the adjacent side / the hypotenuse so if cosine Theta is equal to 3 / 7 Theta is going to be R cosine 3 over 7 and once again you have to use your calculator to figure this out because without a calculator I of you know what this answer is and this is going to be 6462 de so here's another one for you let's say this is five and this is six go ahead and find the value of theta so the hypotenuse is six opposite to Theta is five so we know s is associated with opposite and hypotenuse s Theta is equal to the opposite side which is five divided by the hypotenuse which is six therefore Theta is the arc sign or inverse sign of 5 over 6 and so the angle is going to be 56.4 de and that's it that's all you got to do to find the missing angle of a right triangle for those of you who might be interested in my trigonometry course here's how you can access it so go to udemy.com and once you're there enter into the search box trigonometry now this is a course I've recently created so I haven't finished add in all the sections that I want to add so in time I'm going to do that right now the page is accessible on the uh you can find the course on the second page and here it is trigonometry the unit circle angles and right triangles is basically the one with the dark background and a circle with a triangle inside the circle so let's look at the curriculum in the first section I'm going to go over angles radians how to convert degrees to radians co-terminal angles how to convert DMS to decimal degrees Arc Length area of the sector of a circle linear speed and angular speed work problems and also if you need to take the time that's shown on the clock and if you need to convert it to an angle measure I cover that in this section as well and then at the end of each section is a video quiz the next section is about the unit circle the six trig functions s cosine tangent secant cosecant cant and also reference angles as well after that you have right triangle trigonometry things like so Circle TOA the special right triangles like the 30 60 90 triangle you need to know that so you can evaluate sign and cosine functions without using a unit circle next I'm going to talk about how to solve angle of elevation and depression problems and just solving the missing sides of right triangles after that trigonometric functions of any angle and then graph in trig functions you need to know how to graph the S and cosine functions secant cosecant and uh tangent as well after that inverse trick functions you need to know how to evaluate it and also how to graph it too in addition you need to know how to graph or evaluate compositional trig functions for example you might have S of inverse cosine of 3 over 4 or something like that and you could use a right triangle to uh solve those types of problems you'll see it when you uh access that section after that applications of trig functions solving problems that have two right triangles in it and baronss as well one of the hardest sections in trig is this section verifying trig identities so that's uh that's a hard one so make sure you spend some time learning that section after that sum of difference formulas double angle half angle power reducing formulas product to sum sum to product and also solve in trig equations but there are still some sections I'm going to add to this course like for example law of signs law of cosiness polar coordinates and some other topics as well so about 2third of the course is finished so far and for most students this is just what they need intrigued but in time you'll see more so now you know how to access the course and if you have any questions let me know so thanks for watching