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7.5 & 7.6 Notes

Added Jan 28, 2015 •

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okay guy 7.5 and 7.6 trigonometry we also call it trade for short it only has to do with right angles just like everything else in this chapter we do as we set up ratios looking at the Angles and making ratios out of the site that's called a trigonometric ratio so it's a ratio of the lengths of the sides play right triangle now there's three most common that we're going to use and that sign or sin for short cosine or cos for short engine or tan for short and those are the words that you're going to see on your calculator trig ratios are related to the acute angles right triangle and not the right angle itself so when you're looking at this we're only going to use angle a and angle B here to set up a race no for our definition part we're only going to use angle a sign of a cosine of any intention fake so we're going to set up ratios now if you look at angle itself of course this side is the hypotenuse and will be from angle a you have a side that is opposite of a witch beside a and you have a side that is touching angle at a which is adjacent to it so we'll call it a Jason which is side B so4 sign a the definition it is the side that is opposite over the side that is the hypotenuse so we have opposite over hypotenuse here now that in this case would be side a over side see then we look at the cosine of 8 now the cosine is going to be the Sun that is adjacent to angle a over the hypotenuse again so in this picture that would be cybie oversight see then we look at tangent of a tangent is going to be the side that is opposite over the side is adjacent angle ISO in this picture the opposite is a and the side that is adjacent is be and that's how you're going to set up your ratios in order to this sign is always opposite over hypotenuse  cosine is always the adjacent over the hypotenuse and tangent is always side opposite over the side adjacent now somewhere someway somewhere along the way somebody came up with sohcahtoa to remember this so Soh of so is the sign is opposite over hypotenuse cosine is adjacent over hypotenuse  intangent the TOA is tan is opposite over adjacent and it's just a fun way to remember cosine and tangent sohcahtoa  okay here's an example find the sign of a cosine of a tan of a sign B cosine B 10B Express each ratio as a fraction and a decimal okay so okay so let's do angle a first hears angle a now the first thing you always want to do is identify your hypotenuse so this is your hypotenuse cuz it is opposite of the United green Angle now so for angle a the side that is going to be the opposite of it will be 5 in the adjacent the one that's touching it will be 12 so this will be your opposite and this will be your adjacent and this is only franglais so this what is a which is opposite 5 over hypotenuse 13 the cosine of a will be the adjacent 12 over the hypotenuse 13 and the tangent of a will be the opposite 5 or the adjacent 12 now if we look at angle B I'll do it a different color angle B this is still our hypotenuse but the side opposite this time going to be 12 in the adjacent will be 5 so I will write that and blue so this time this is the opposite and this is the adjacent for side for angle be so my sign of angle B is opposite is 12 and hypotenuse is cosine of the adjacent to be as 5 and the hypotenuse is 13 and the tangent of the is opposite 12 over adjacent fight it's a 12 now you can turn those all into decimals also but if you I just want you to notice something that the cosine of angle a is the same ratio is this sea and likewise the sign of a is the same ratio is the cosine of B and in the tangents are just reciprocals of each other  now too are you a trig ratios you can use a scientific calculator or a trig chart when using a calculator you must make sure that your calculator is in degrees and radiance  so you can do that by hitting the mode button and making sure the degrees is highlighted and not the radiance  now look at the did I gave you that was the tree chart and look under there and find angle 49 degrees and then look at what the sign is in there should be a long decimal and it should play .75 for 7 and that's how you find it on a tree chart then for this one for B you would look up and find angle of 8 and then go over to the tan column and it should be 7.115 for and that's what the tan of 82 degrees should equal now the other way is to type it in your calculator so on your calculator if you will look and hit the button that says cosine what type in three degrees in hitter you should get point 99864 this one hit the sign button and type in 87 degrees and hit enter and you should get point 9986 and the reason is 3 degrees and 87 raise equal 90 degrees so they are opposites of each other that's why they're going to equal the same thing cuz one is cosine and one is sign okay so this is how we put it all together and use those ratios to actually saw for the side now if you notice they give me the measure of angle a right here 36 degrees now identify your hypotenuse first and then you won't ever have any problems so this is my hypotenuse okay now if I'm using angle a the side that is opposite of a is X and a side that is a Jason is white so this is my opposite and this is my Jason site now have to look at what you're giving and decide which one of those ratios to use now if we want to solve for x first X's our side that is opposite we have the hypotenuse which ratio has the opposite and hypotenuse in you'll find out that that is sign so if I do the sign and it said of angle a you actually put in the degree 36 degrees equals x over the easy way to solve this to put this over one in cross multiply so one x axis x equals sine of 36 x 8 now I think it's easier to put the a front of the sign of 36° know what you do is you will type this in your calculator 8 x you'll hit the sign button can you hit 36 and hit inner you should get that X is equal to 4.70 to 4 now if you not get that that means your calculator could be in radians and you need to go back and hit that mode button and change it back to the grease okay no that's how we found fun why now why is the side that is adjacent so we have the hypotenuse in the adjacent side so if you look at your ratios who sign is what is adjacent over hypotenuse so if I do cosine of angle a which is 36 degrees that will eat the adjacent why over the hypotenuse 8 again put that over one cross-multiply so that's why equals and again put the number in front 8 x cosine of 36 degrees in your calculator and you should get 6.47 to 1 so if we shorten is 4.7 this side is 4.7 and if we shorten this turn that would be 6.5 so why is equal to 6.5 and that's how you do those problem okay now let's look at it I noticed that we know this angle is 34 in this thing was 56 so we can use either angle and it doesn't matter which one you pick so let's pick the 34 so this would be the side that is opposite and this would be the side that it is adjacent here's our hypotenuse and we don't know anything about it so opposite and adjacent is tangent so if we use our tan we do tan of the angle that we decided to you 34 is equal to the opposite X over The adjacent 12 put that over one cross multiply men's exercise and then tan 34 * 12 - 12 times the tangent of 34 type that in your calculator 12 X and Y 34 and you should get x equals 8.09 for one and that is your answer so this side right here is .1 if we're out again now if you needed to find the hypotenuse if you just think about it you have two sides now you can actually use the Pythagorean theorem to find cat's eye  okay soon light orgy they give us the 50° but if you think about it if that's 50 than this one has to be 40 to make an degree cuz that one is not so to make a right triangle so you can actually use either one it doesn't matter now if I wanted to keep with it I need to find the X and the Y so let's say that I choose to use the 50 degrees angle so if I'm going to do x x is the side is opposite and the side that is given is adjacent so opposite and adjacent would be tangent so I would do tan of the angle I chose 50 degree angle the opposite of that is x in the adjacent is 5 again put that over one cross multiply you get x equals 5 times the tangent of 50 degrees top 10 in your calculator exactly like it looks and you will get x equals 5.9 9  play now if I want to find why why is the hypotenuse  so again I can still use my 50 degrees angle or you can switch to the 40 it doesn't matter let's use the adjacent and the hypotenuse now that is cosine so the cosine of the angle 50° that is equal to the adjacent 5 over the hypotenuse why now put that over one in cross multiply look what happens this time 1 * 5 is 5 equals y times the cosine of 50 degrees now remember saw for why so what you would do is you and divided by the cosine of 50 degrees so those cancel and then over here you would do cosine of 50 degrees so in your circulator you would type 5/8 cosine and then type in 50 hit enter and you would get y equals 7.77 6  and that's what you would get so if you go back up here if we round this size going to be 5.6 and this what will be 7.8 so you have 5 5.6 and 7.8 for your sides down letter H we only have to find X which is across from the 90° so this is our hypotenuse okay now you can use either the 10 degree angle or the 80 degree angle let's use the 10 degree angle they give us the side that is adjacent to the 10 degree angle so we have a Jason and hypotenuse that would end up being our coach sent so cosine of 10 please what equal that you and Jason side 16 over the hypotenuse X put that over one cross multiply and again we get the same thing 16 equals x times the cosine of 10 degrees and we want to get X bites self so divided by the cosine of 10 degrees on both side these will cancel and in your calculator you'll just type in 16 / cosine of 10 and you'll get X is equal to 6 18.25 and that is your answering your hypotenuse and again from there you could actually use the Pythagorean theorem to find the third side that you don't know I need to make one correction on G I noticed that when I rounded right here on this raise up at 5.6 actually that would Round Up to just a 6 so here would be 6 not 5 Points

Uploaded by Pam Chastain.

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7.5 & 7.6 Notes

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