 Hello friends, so we are going to discuss a new topic and this is Limits to the value of t ratios that is trigonometric ratios. So we are going to understand What are the values or the range of values all the six t ratios can take? So start with this for this identity sine square theta plus cos square theta is one, right? so we are first trying to attain or arrive at the range the values of sine theta can take So sine square theta plus cos square theta is one now You know that sine square theta is greater than equal to zero all the time. Why? Because it is a positive point. It's a square term So square of anything if you square anything it was always going to be greater than equal to zero if it is zero Then square of zero is zero But if it is anything negative or positive square of any anything any number will be always positive So I can say sine square theta is greater than equal to zero. Similarly, cos square theta will always be greater than equal to zero Now some of two so two positive quantities are there one is sine square theta another is cos square theta and some of these two is one right that means one Two positive quantities are added up to get one That means each one of them will be less than one so hence sine square theta is less than equal to one and Cos square theta is less than equal to one So this will be one. So let's say when sine square theta is one then definitely cos square theta is zero and When cos square is one then definitely sine square theta is zero. So this is what it means So now sine square theta is less than one So anything which is less than one that means if the square of anything is less than equal to one It always means that the value will be from minus one to plus one Is it it you can take examples for example if it is a zero point two negative So if you square this you will get 0.04 right and if it is let us say 0.3 so square this you will get 0.09. So if you see These all are greater than equal to zero. So till let us say if you take minus 0.99 999 like that. So it's square again will be equal to very near to one again So 0.9 only 99 like that Okay, so you you can check similarly minus one will be less than cos theta and less than equal to one Right. So hence we got the first two ranges. So sine theta is less than equal to one and cos theta is less than equal to One and sine theta is greater than equal to minus one and cos theta is also greater than equal to minus one This is the first Result now Cosic and theta is nothing but one upon sine theta reciprocal of sine theta So hence when sine theta is less than one then one upon sine theta will always be greater than equal to one Right. So if any anything is less than one less than equal to one reciprocal of it will always be greater than that Is it it so hence if you see Cosic theta is greater than equal to one you can check taking value Let us say point two is less than one but one upon point two will always be greater than one Correct. So this is for co-second theta. Similarly, if you if you take secant theta is a one upon cos theta So hence secant theta will always be less than equal to minus one and secant theta will be greater than equal to one So cosecant and secant in terms of range Are similar that means both cosecant and secant have same range that means It will be either less than minus one or it will be greater than one Correct. Now, let's let's talk about tan theta So tan theta is sine upon cos which when which can be written as under root one minus cos square theta by cos How because sine square theta plus cos square theta is equal to one So hence sine theta sine square theta can be written as one minus cos square Theta. So hence sine theta can be then reduced to this one minus cos square theta. Is it it? So hence I have represented sine in terms of cos itself. So one minus cos square theta under root divided by cos theta Now look at this particular term carefully. Now, it means cos theta tends to zero from the positive side What does it mean? So this arrow means tends to so when cos theta value becomes very very close to zero But the value is positive now The value can be either positive or negative. Let us say this is my x-axis and let us say in x-axis I am representing cos theta values So cos theta well, let us say this is my one and this is minus one So cos theta value in this area is less than one But it is positive similarly cos theta value here in this area this region is Negative and but very close to zero, isn't it? So hence cos theta value can be let us say point plus point plus point zero point zero zero zero one and it can also be minus zero point zero zero zero one Right. So when I'm talking about zero plus I mean to say that cos is positive but very close to one and when I say zero minus I say cos is very close to zero, but it is a negative number, right? Now as cos is very close to zero, but not zero mind you. It's very close to zero, but not zero So tan theta takes a very large positive value. Why because I am dividing by a very very small positive value So let us say if you divide one by zero point zero zero zero zero and many zeros Right followed by let's say more lots of zeros and one then this is a very huge number, isn't it? It will be ten to the power number of zeros here, right? It let's say 20. So it's a very huge number So hence if you divide anything by a very very small positive number then the number and You know the quotient or whatever is the result of the division is a very very large positive value, correct? So that means tan theta is taking a very large positive value It can take any, you know positive value and in that sense we say that tan theta tends to positive infinity similarly When cos theta is very very small Value but negative value, right? Then tan theta will take a very negative large negative value Why because let's say the same thing here if I change this changes to minus one then this Item becomes minus 10 to the power. Let's say 20 20 is just a number. I'm saying it's a very large negative value So what do we learn? We learn that tan theta value can be Now tan theta can definitely be zero. Why because if my sine theta becomes zero then tan theta becomes zero Okay, so hence what happens? I can see that tan theta is taking any value between minus infinity to plus infinity, right? Then theta range is between minus infinity to plus infinity similarly cot cot theta also can be explained in the same terms and Again, you can take two values of sine very small positive and very small negative and you will see cot theta like tan is also the same minus infinity to Plus infinity. So what is the summary? So I have summarized the above results if you see So basically it is nothing but sine theta will be varying from minus one to one and it in Alternatively, it can be also said that mod value of sine theta is less than equal to one, right? The modulus whenever you take the mod value of sine theta, it will always be less than equal to One and yes, you can always write this as well. So zero less than equal to Zero less than equal to sine theta mod sine theta is less than one because mod is a positive quantity So it will not go into the negative realm. So hence. This is another way of writing this Right and similarly for cos is this so minus one less than equal to cos theta less than equal to one minus infinity less than tan theta is less than infinity here if you notice I have not put equal to sine because you cannot equate anything to infinity because you know, it's in a way It's you know, I'm not I'm not known not defined number. So hence I am I cannot equate it to tan theta It is it is to it is to show that the infinite is a very large number. It's not equal to tan theta Okay, similarly minus infinity is Is less than equal to quarter theta, sorry, less than quarter theta and less than infinity Seq and theta again less than minus one and see can theta is greater than one Similarly, cosecant theta is less than minus one and cosecant theta is greater than one, which can be again written like this Hope you understood this this information is very vital We'll also have a demonstration video on which we'll show you how the graph of Sine theta versus theta behaves or all the trigonometric ratios with respect to the angle behaves You can check that out in the same Channel