hey class we're going to do uh part two on special right triangles here we're going to start out with some calculator buttons so um couple things to note when you're putting in trig functions into your calculators if you are given degrees remember to make sure that your calculator is in degree mode so degrees should be highlighted different calculators look different so make sure you know how yours works now for most calculators they only have the three main trig functions s cosine and tangent so if you want to find the cosine of 37° you just type it in cosine of 37° which is 0.7 986 we're going to go to four decimal places however you'll notice my calculator does not have a camp button some calculators do the majority do not so what we have to remember with secant is that it is the reciprocal that secant of an angle is defined to be 1 / cosine of that angle and so if you want to find the secant of an angle and that's not a right triangle you have to type it in as 1 / cosine of that angle and so if you do 1 / cosine of 54 oops that was not right 1 / cosine of 54 that will give you the correct value for the secant of that angle now notice that when we did the cosine of an angle it is less than one we got 7986 that makes sense because remember you find cosine by taking the adjacent side over the hypotenuse of a triangle and then cant is above one because that is the hypotenuse over adjacent and the hypotenuse is always the biggest side of a triangle and so the coine of an angle should always be less than one because the adjacent side is always smaller than the hypotenuse and the cant should always be more than one because you're going to have the H the bigger side on top so something to keep in mind when you're plugging them into your calculator to double check if they're correct so if you want tangent of 12° you can use your tangent button but if you want coent so 0 21 26 but if you want coent you have to do 1 / tangent of that angle so 1 / by tangent of 72 would be 03249 so go ahead and practice those get familiar with your calculator so you know where those buttons are the next thing we're going to talk about in this unit is the special right triangles and special right triangles include two we have the 45 4590 triangle which is the Isles triangle so since the angles are congruent the sides opposite the angles are congruent congruent and the ratio for this triangle is 1 1 < TK 2 which means that the side lengths are the same and the hypotenuse is square root of as two as square root of two times as big so um in a standard 1 1 < TK of 2 or it could be 2 2 2 roots of 2 or it could be 5 5 five five square roots of two so the hypotenuse is our square root S 2 times as big as the legs and the legs are always equal so we'll see how that works you have a nice equation here you can use two and then we have the 306090 triangle now the 306090 triangle has a 30° angle which is the smallest angle the smallest angle is across from the shortest side because the Angles and side lengths are related and then we have a 60° angle and that's going to be across from the longer side and then the 90° angle is the biggest angle across from the longest side which is the hypotenuse now this has a ratio of 1 square roots of 3 2 which means that the short leg the short leg and the hypotenuse the hypotenuse is two times as big as the short leg so hypotenuse is twice the value of the short leg and then the long leg is the sare < TK of 3 times as big as the short leg so if you have a short leg is X then you would take 2 * X to find the hypotenuse and then you would take that x value time the < TK of 3 to find the long leg all right let's do a few examples so this is our 454590 triangle I'm given a side length which is five since the angles are the same the sides are congruent and the ratio is 1 1 squ roots of two which means that if this is five five this would be five squ roots of two for the hypotenuse to keep that ratio the same in the 30 6090 triangle example here I'm given the hypotenuse the hypotenuse is the longest side the ratio is 1 < TK 3 2 if your longest side is your two that means your shortest side is half of that so you can also remember it as x x s roots of 3 and 2x so your shortest side is going to be half of your hypotenuse so six and then your long side is going to be the sare < TK of 3 times as big as the short side or 6 square roots of 3 so that the ratio is proportional it's all six times as big so the values I was looking for here was the the short side and the long side okay to do a couple more of these we have a 45 4590 it's kind of tipped a funny way but this is my hypotenuse so these sides are congruent so this would also be 9qu roots of two and your ratio is 1 1 roots of 2 which means if your side length is 9 roots of 2 then this side is also 9 roots of 2 then your hypotenuse is going to be 9 roots of 2 * another < TK of two because the hypotenuse is our square otk of two times as big if it's like a 1.414 so 1.4 times as big as the leg and then to multiply this you'd get 9 * the < TK of 4 which is 9 * two so your hypotenuse is going to be 18 there this one is similar to our last example with the 30 60 90 the 8 is across from the right angle so that is our hypotenuse it's 1 < tunk 3 2 for this one and it goes in order because Square < TK of 3 is like 1.8 so 1.7 so it's 1 1.7 two so the two legs are just it goes to the biggest so the hypotenuse is double your short leg so if the hypotenuse is eight the short leg is four and then the long leg across from the 60 is the square < TK of three times as big as the short leg or 1.7 times as big so we'd leave that as four square root of 3 4 * theun of 3 all right we're going to do it a few more times so if you have solving a triangle remember that's finding all of your U missing side lengths if you have a 30° angle we know that the other angle has to be 60 because it has to add up to 180 and 30 and 60 make 90 and plus another 90 is 180 and then since it's a special right triangle we want to find our exact answer so instead of using trig we're going to use our special right triangle ratios one thing I noticed is that we are given the long leg here so my ratio for this triangle is 1 < TK 3 2 and remember that goes in order 30 60 90 from smallest to biggest 1 1.72 and you're given seven as the side across from the 60° so if you add some x's in here like if this is 1 X and this would be x * < TK of 3 this would be 2 * * X so your ratio would be x xun 3 and 2x that means that your long leg here of s is your x * < TK of 3 so your short leg * the < TK of 3 gave you s so to find your short leg you have to work backwards so since the short leg times the < TK of 3 equals 7 you would take 7 ided theare < TK of 3 to find that value and go backwards so that means that your short leg would be 7 over the < TK 3 that could give you a decimal answer as a exact answer you're going to want to rationalize that so you're going to leave it as 7 OTS 3 over 3 so that's going to be your short leg which is our a value and then so this is 7un 3 over 3 and then our hypotenuse is going to be double that oops over three and so hypotenuse is going to be 7 Roots 3 over 3 * 2 because it's double and remember you take a fraction times a whole number the whole number only applies to the top so 7 * 7 Ro of 3 * 2 would be 14 roots of 3 all over 2 which is going to simplify to just 7 squ roots of 3 so there is our right triangle solved so that one was a little bit tricky because you were given the long side and you had to work backwards to find the short side which means you had to divide by the < TK of 3 instead of times by the < TK of 3 and then one more here using um special right triangles we have a ship that travels 53 mph Northeast on a course that forms a 30° angle with the Positive xaxis Okay so so the ship is traveling positive xais 30° angle so it's traveling north you know never eat soggy waffles it's traveling Northeast a 30° angle it is traveling 53 M hour for 2 hours and so if it's traveling 53 miles oops 53 miles per hour how far is it going in 2 hours you would have to multiply these together so 53 * 2 it's going to go a total of 106 miles in 2 hours that's going to be this distance right here 106 miles that's the value or that's the course of this ship then it's asking you to find how far east and how far north typo there but how far east and how far north so this we drop it down into a right triangle this is our North distance and then this would be our East distance and it's a 30 6090 triangle and we're given the hypotenuse so we can solve this using our ratios we have a ratio of 30 60 90 is 1 < TK 3 2 our 90° side is 106 which means that half of that would be 53 and that would be our North distance because that is across from our 30° angle and then we'd have 53 * the sare < TK of 3 would be our East distance because that's across from our 60° angle so that'd be our exact answer if we wanted to approximate that it would be 53 roots of 3 or approximately 91.8 miles so we could answer this question by saying that it traveled um 91.8 miles east and 53 miles north and that is it for special right triangles it does take some practice to get this down so keep practice see and thanks for watching