 Hello friends, so welcome to another session on trigonometry and today we are going to understand Complementary angles and not only we are going to define and understand complementary angles We are also going to understand the rich other relation between various t ratios of complementary angles Now let us start this session by understanding what complementary angles are Now two angles are said to be complementary when their sum is equal to a right angle That is if two angles are there theta and phi and if their sum is 90 degrees Then we say that theta and phi are complementary to each other theta and phi are complementary angles Complementary angles. Okay. This is a very vital information. Keep this in your mind. Now examples 60 and 30 sums to 90. So hence it is They are complementary to each other similarly 40 and 50 degrees are complementary angles Here also if you see 10 and 80 70 and 20 all these pairs are of complementary angles now similarly 12 degrees, let us say and 78 degrees these are complementary angles 12.5 degrees and 77.5 degrees these are all so complementary so you can find out infinitely Many pairs of complementary angles now if you notice in a right angle triangle The two angles apart from the 90 degree are always complementary right in a right angle triangle The other two angles apart from the right angle are always complementary Why because if one is theta by angle some property, we know that it is 90 minus theta, right? So here if you see instead of this, yeah, right angle be here. So if you see Angle 90 degree plus theta which is angle C here plus angle B is 180 degrees. So hence angle B is 90 minus theta right by angle some property now, let us Analyze the t ratios of complementary angles. What does it mean? So let us first first find out sine of theta So if you see sine of theta is nothing but by definition its opposite side to the theta That is this side AB divided by what the hypotenuse that is this side that is BC So hence sine of theta is AB upon BC Similarly, if you look closely cos of beta now if you see from this angle 90 degree here 90 minus theta if you see from this Perspective so cost of this angle is nothing but the adjacent which is AB which is highlighted over here This is AB this one divided by what this line, right? This is cos of 90 So hence if you if I have written here So both sine of theta as well as cos of 90 minus theta are same ratio that is AB by BC So hence I can write sine theta is equal to cos of 90 minus theta This is a very vital information and this is going to be used by you if you pursue trigonometry In your career very very important information now Similarly, you can see cos of theta if you see cos of theta is nothing but this side now in this case This one this divided by The hypotenuse right that is AC by BC which happens to be the sine value of 90 minus theta Why because for 90 minus theta this side is opposite isn't it and this one is hypotenuse so hence What do we see cos of theta is? Equal to sine of 90 minus theta very important information Similarly other T ratios also cosecant now another important one is tan theta again So tan theta is nothing but what is tan theta in this case Opposite by Adjacent so AB by BC and hence which is nothing but cot of 90 minus theta so tan theta and cot 90 minus theta are Related they are equal now other is just a formality other T ratios cosic cosecant theta is 1 upon sine So if you go by whatever we found out sine is cos of 90 minus theta So that is secant 90 minus theta isn't it you can also do it by the Definitions of the T ratios that is in terms of the ratios of the sides if you want to calculate You will get the same value. So please remember cosecant is secant 90 minus theta now secant theta is also equal to cosecant 90 minus theta right now this is what we Derived and now let us write down the summary the summary is sine of theta is cosine of 90 minus theta now Don't think that this is only restricted to a right-angle triangle. Whatever is the value of theta? Cos of 90 minus theta will be Same to sign theta. Let us take an example also. So for example in the previous sessions we learned Sine of 30 degrees is nothing but half if you if you have seen those sessions lectures So sine 30 is half and now sine 30 degrees can be written as sine of 90 degree minus 60 degree isn't it now from the relation sine of 90 minus theta if you see is cos of theta So hence I can write this as cos of 60 degrees now if you see cos 60 also is half Isn't it? Let us take another example. Let us say tan of 60 degrees now tan of 60 degrees can be written as tan of 90 degrees minus 30 degrees right now Let us see what is tan 90 minus theta than 90 minus theta is cot theta. So that means this will become cot of 30 degrees Isn't it now what is tan 60 by the table if you see it is root 3 if you have the table with you You can check tan 60 is root 3 and similarly cot 30 also is root 3 root 3 right That means our whatever information or results we derived are accurate, right? It is true for all values of theta. So, please remember these relationships, right? So sine and cos are complementary if you see cos and sine same thing tan and cot tan and cot and secant and cosecant, right? So if one is theta the other will be 90 minus theta value I hope you understood these concepts will take up more problem solving on these concepts in problem solving sessions. Thank you