 hello friends so continuing with our sessions on trigonometry in this session we are going to illustrate how the trigonometric ratios though they depend on theta but they do not depend on the sides or you know the precise length of the sides of a triangle so since it is a ratio so hence if we change the value of the sides of the triangle the ratios do not change yeah so this particular illustration is to describe that so far the theta is constant or the angle is constant the trigonometric ratios are not going to vary they are not going to be dependent on the length of the sides of the triangle so what we have done is we have shown two triangles here one is triangle a b c and another triangle is a d e okay and a b happens to be the radius of the outer circle and this circle is the is another concentric circle to the outer one and a d happens to be the radius of the inner circle we have dropped the perpendicular from b on to x axis like this b c and d e is also perpendicular to x axis so we get two right triangles here one right angle a b c at right angle at c another right triangle at right triangle a d e right angle at e so you now see as i will move the point b the angle would change angle would change but the ratio in both the triangles ratio of so we have taken here sin theta only one of the six ratios you can check all other ratios will behave similarly so sin theta value in both these triangles will stay the same despite the fact that their individual side lengths keep changing so if you can see a b is of you uh length one uh so i'll just take it here yeah so this is h equals to one is the length of a b h one is ad which is point five similarly b one is shown here as point three one which is a e a e is nothing but b one and ac is point seven one you can see similarly dc is point seven one and ad or d e is point three five now as i will change the position of b you'll see the values of all these sides would change only the hypotenuse that is the radius will not change but base and perpendicular values will change but despite that the sin theta value is not going to change you can see one of the cases is mentioned over here in this case theta that is 45 degrees it is given as 45 degrees here so at 45 degrees p by h in the outer triangle is point seven one upon one which is point seven one and sin theta for the inner triangle also is p one upon h one now p one was point three five and h one is point five so if you see both the sides are exactly half of the sides given in triangle a bc so hence the ratio doesn't change now i'm going to vary this angle by changing the position of b and you will see that the ratio that sin theta value doesn't change so as i'm increasing the value of sin theta or theta you can see eta is now going towards 90 degree but the sin theta remains the same in both the triangles isn't it so if you see both the triangles are having the same value of sin theta now if you can see as i'm decreasing the value of theta then also the sin theta value is not changing both in both the triangle sin theta value stays the same so you could have figured out by now that the sides of the triangle are varying proportionately isn't it you will learn this concept in similar triangles as well you can use the concepts of similar triangle as well to prove that the ratio of the two sides which are mentioned here as p by h and p one by h one will not change if because the triangles are similar in every case isn't it so this demonstration was to just describe to you that sin theta value doesn't change even if it is in different if even if the theta is in different triangles likewise all other t ratios will not change till the value of theta is same whether it is in smaller triangle or any other bigger triangle