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Video: Trigonometric Ratios for Complementary Angles

Lauren McNaughten

Use the nonright angles in a right triangle to identify the relationships that exist between the trigonometric ratios for complementary angles and apply this knowledge to solve problems without using a calculator.

09:08

Video Transcript

in this video we're going to look at the special relationships that exist between the trigonometric ratios for complementary angles so I'm starting off with a right-angled triangle in which side label the three sizes x y and z I haven't specified any units here because it doesn't particularly matter but we were seeing you all the same so we'll centimetres or inches or whatever I've also label the two angles that aren't right angles as alpha and beta no alpha and beta are the complementary angles that were interested in because a member complementary angles the definition is that they sum to ninety degrees and of course as this is a triangle the total sum of its angles is one hundred and eighty° as the other angle is a right angle ninety° that means that there's ninety° left over for alpha and beta and therefore their son must be ninety so alpha and beta are the complementary angles that we interested in so what I need to do if I'm just going to work with each of these angles in ten I'm going to start with angle alpha and first and then a label the three sides of a triangle in relation to angle alpha so the hypotenuse the opposite and the adjacent hypotenuse of a right-angled triangle is a course always the same side but the member than the adjacent in the opposite change depending on which angle you interested in so in relation to angle alpha why is the opposite annexes the adjacent now we're going to do is I'm going to write down the three trigonometric ratios sine cosine and tangent for angle Alpha so signed festival sign member is opposite divided by hypotenuse so that will be y divided by Z cosine is adjacent divided by hypotenuse so that will be x divided by Z finally tangent is opposite divided by Jason so that will be y divided by x if you struggle to recall any of these definitions of sine cosine or tangent then you need to remember that word socket hour in order to help you with that so socket error remember takes the first letter of each of these words in the case of sign for example sign is opposite divided by a posh news and that's where the song comes from right now I'm going to do exactly the same thing but with angle beta so I'm going to leave all the sides again but in relation to angle beta this time for hypotenuse as we said is always the hypotenuse but this time for angle beta the opposite is this side here side x and the adjacent is this side her side why and I just like I did for angle alpha I'm going to write down the three trigonometric ratios for angle beta in terms of x y and z bofour beta sign of beat it is opposite over hypotenuse so looking at the orange labels this time that will be x divided by Z fourK sine cosine is adjacent divided by hypotenuse so again looking at the orange labels that will be y divided by Z finally tangent is opposite divided by a Jason so looking at the orange labels that will be x divided by y so I have the three trigonometric ratios sine cosine and tangent written down for both alpha and beta this pair of complementary angles in terms of x y and z now if you just have a look at these different trigonometric ratios there a couple of things that will become apparent if you look at sign of Alpha that is equal to y over Z and what your notice on the right hand side of the screen but we also have why over said again but this time is equal to Cos or cosine of beta so what this tells us then is that sign of angle alpha must be equal to cosine of angle beta so what we having is that sign of Alpha is equal to cause of beta no remember that alpha and beta are complementary angles so we had at top of the screen that alpha beta is equal to ninety° we could also write this down alternatively as beta is equal to ninety - Alpha and so if I replace beta with this expression in terms of Alpha then it will give me a general relationship so I have this relationship here which is that sign of any angle alpha is equal to cosine of ninety - that angle that relationship will always hold true for complementary angles now looking back at these ratios you'll also see that there are two places where it's over said it is so cause of Alpha is equal to Exeter z and sign of beta is also equal to x over Z so therefore I must have that cos of Alpha is equal to sign of beta again if I replace beta with ninety - alpha then it gives me another general relationship for complementary angles so I have that cause of Alpha is equal to sign of ninety - Alpha fine if you look at tangent we have town of outfit is wirex and turn of beta is x over y so that these two tangents are reciprocals of each other because the fractions are just inverted in each case so the relationship here is the town of Alpha is equal to one/ten of beta and again if I then write this relationship down where I replace beta with ninety - Alpha then I have the general relationship that time of Alpha is equal to one over ten of ninety - Alpha so where you have a pair of complementary angles that is angles that sum to ninety degrees these three relationships will exist between the values of their trigonometric ratios so between their sine cosine and tangent and that's due to the fact that the opposite and the adjacent sides in a right-angled triangle swap round for the two complementary angles ok let's just see how to use with just worked out in order to apply to a question so we giving a table of information in which were given the values of the sine cosine and tangent ratios for two angles twenty degrees and sixty degrees and we asked to use this information to write down cause of thirty and sign of seventy of course you assuming in this question that we haven't got access to a calculator so we just need to use the information in the table and nothing else so thinking about what we just did and complementary angles where what was spotted us something about thirty° but were given some information about sixty degrees and their complementary angles because they some two hundred and ninety the same thing with the seventy degrees and twenty degrees so let's think about cause of thirty festival we saw in the previous page that cos of an angle alpha is equal to sign of ninety - that angle so I can use this to write down something about cosine of thirty and sine of ninety - thirty so I have it cos of thirty is equal to sign of ninety - thirty-nine cos ninety - thirty is just sixty so I have that cause of thirty is equal to sign of sixty and using the information in the table I can look that value up and see that it's equal to root three over two that's an exact that you as I said so I've just used the relationship that exists between sine and cosine for this pair of complementary angles and then information in the table in order to work that out indeed if you do have a calculator you can then confirm that it is in fact the correct value kids method for the second one is very similar so sine of seventy degrees while a member sign of Alpha is equal to cause of ninety - Alpha so we can use a very similar process sign of seventy will because of ninety - seventy or cos of twenty and then if I let this value up in the table cos of twenty to three significant figures is zero point ninety-four hundredth so in summary then we've demonstrated the relationship that exists between the trigonometric ratios for complementary angles and then we've seen the type of question where you may need to apply this knowledge