have you ever heard of the expression sohcahtoa 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 of 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 squared plus b squared is equal to c squared but 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 sine cosine tangent opposite adjacent hypotenuse sine theta according to sohcahtoa s is for sine o is for opposite h is for hypotenuse sine theta is equal to the opposite side divided by the hypotenuse cosine theta is equal to the adjacent side divided by the hypotenuse k is for cosine is adjacent over hypotenuse and tangent theta toa is equal to the opposite side divided by the adjacent side so that's the tangent ratio it's opposite over adjacent now we know that cosecant is one over sine so cosecant is basically hypotenuse divided by the opposite side you just need to flip this particular fraction secant is the reciprocal of cosine so secant is going to be hypotenuse divided by the adjacent side cotangent is the reciprocal of tangent so if tangent is opposite over adjacent cotangent is adjacent divided 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 the missing side of this right triangle and then find the values of all six trigonometric functions sine cosine tangent secant cosecant cotangent now to find the missing side we need to use the pythagorean theorem a squared plus b squared is equal to c squared so a is three b is four and we gotta find missing side c which is the hypotenuse three squared is nine four squared is sixteen nine plus sixteen is 25 and if you take the square root of both sides you can see that the hypotenuse is 5. now it turns out that there are some special numbers there's the three four five 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 2 we'll get 6 8 10. that can also work or if you multiply by 3 you get the 9 12 15 triangle if you multiply this one by 2 you get the 10 24 26 triangle those are also special triplets they work with any right triangle now some other numbers that are less common but you might see are the 9 40 41 triangle and the 1160 61. so if you see some of these numbers you can find the missing side quickly if you know them so now let's finish this problem so what is the value of sine theta so according to sohcahtoa we know that sine theta is equal to the opposite side divided by the adjacent side and the part so soh opposite to theta is 4. and hypotenuse is five so therefore sine theta is going to be four divided by five now cosine theta is equal to the adjacent side divided by the hypotenuse we said 4 is the opposite side 5 is the hypotenuse and 3 is the adjacent side so in this case is going to be 3 divided by 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 divided by 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 one over sine so just flip this fraction is going to be five over four and secant is the reciprocal of cosine so flip this fraction secant is going to be five over three cotangent is the reciprocal of tangent so if cotan i mean if tangent's four over three cotangent is going to be three over four 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 missing 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 squared is 64 and 17 squared is 289 289 minus 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 sine theta cosine theta tangent theta and then cosecant theta secant theta and cotangent theta so using sohcahtoa we know that sine 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 divided by 17. so that is the value of sine theta now cosine theta is going to be equal to the adjacent side divided by the hypotenuse so the adjacent side is 8 they have hot news 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 is 8. therefore tangent is going to be 15 divided by eight now cosecant is the reciprocal of sine so if sine theta is 15 over 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 over 8. you just got to flip it and cotan is a reciprocal of tangent so cotangent 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 gotta 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 three times five is fifteen four times five is twenty five times five is twenty five so we have the fifteen and we have the twenty five therefore the missing side must be twenty 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 sine 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 sine theta we know it's opposite divided by hypotenuse so it's 20 over 25 which reduces to 4 over 5. if we divide both numbers by 5. 20 divided by 5 is 4 25 divided by 5 is 5. cosine theta is adjacent over hypotenuse so that's 15 divided by 25 which reduces to 3 divided by 5. tangent theta is opposite over adjacent so 20 over 15 which becomes if you divide by 5 that's going to be 4 over 3. now cosecant is the reciprocal of sine 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 divided by 3. now if tangent is 4 over 3 cotangent has to be 3 divided by 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 degrees and this side is 42. so what trig function should you use in order to find the value of x should we use sine 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 degrees is equal to the opposite side x divided by 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 0.7813 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 degrees and we're looking for the value of x and hypotenuse is 26 which trig function should we use sine cosine of 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 divided by 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 degrees cosine 54 is 0.587785 if we multiply that by 26 this will give us the value of x which is 15.28 here's another one that we could try let's say the angle is 32 degrees and the hypotenuse is x and this is 12. so notice that 12 is opposite to 32 and we have the hypotenuse so this time we need to use the sine function sine of the angle 32 is equal to the opposite side 12 divided by 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 times 12 is 12 and this is going to equal x times sine 32. next i recommend dividing both sides by sine thirty-two sine thirty-two divided by itself is one so therefore x is equal to 12 over sine 32. 12 divided by sine 32 is 22.64 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 5 is opposite to the angle and 4 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 divided by the adjacent side 4. so how can we find the value of the angle theta if tangent theta is 5 over 4 and then theta is going to be the inverse tangent or arc tangent of 5 over 4 and you simply have to type this in your calculator so type in arc tan 5 over 4 and you should get an angle of 51.34 degrees so that's how you could find the 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 associative of cosine cosine theta is equal to the adjacent side divided by the hypotenuse so if cosine theta is equal to 3 divided by 7 theta is going to be arc cosine 3 over 7. and once again you have to use a calculator to figure this out because without a calculator out of you know what this answer is and this is going to be 64.62 degrees so here's another one for you let's say this is 5 and this is 6. go ahead and find the value of theta so the hypotenuse is 6 opposites of theta is five so we know sine is associated with opposite and hypotenuse sine theta is equal to the opposite side which is five divided by the hypotenuse which is six therefore theta is the arc sine or inverse sine of five over six and so the angle is going to be 56.44 degrees 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 adding all the sections that i want to add so anytime 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 coterminal angles how to convert dms to decimal degrees arc length area of the sector of a circle linear speed and angular speed word 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 the video quiz the next section is about the unit circle the six trig functions sine cosine tangent secant cosecant cotan and also reference angles as well after that you have right triangle trigonometry things like sohcahtoa the special right triangles like the 30-60-90 triangle you need to know that so you can evaluate sine and cosine functions without using the 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 the graph intrigue functions you need to know how to graph the sine and cosine functions secant cosecant and tangent as well after that inverse trig 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 composition of trig functions for example we might have sine of inverse cosine of 3 over 4 or something like that and you can use a right triangle to solve those types of problems you'll see when you access that section after that applications of trig functions solving problems to have two right triangles in it and barons as well one of the hardest actions 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 summer difference formulas double angle half angle power reducing formulas product to sum sum to product and also solve and trig equations but there are still some sections i'm going to add to this course like for example law of sines law of cosines polar coordinates and some other topics as well so about two-thirds 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 you