 Hello friends. So welcome back to the session on finding t ratios of some specific angles Today, we are going to learn to find out the t ratios for theta equals to 60 degrees I'm in a triangle ABD. It's given that angle a is 60 degrees. So it's 60 degrees here. It's mentioned angle DAD is 60 degrees and Angle D is 90 degree, right? So now we have to find out The values of different t ratio. We have to find out what is sine 60 what is cos 60 degrees and so on and so forth Okay, now let us see how to find this out Consider a triangle ABC such that ABC is an equilateral triangle, right? So that AB is equal to a BC is equal to a and AC is equal to a right all the three sides are a of an equilateral triangle now Obviously angle all the three angles of equilateral triangle are also equal, which is equal to 60 degrees So we can say in triangle a BD Angle DAD is 60 degrees. Okay, and angle B D a is 90 degrees. So basically BD is nothing but a perpendicular dropped from one of the vertices of the equilateral triangle onto the side AC So BD is perpendicular to AC now if you drop a perpendicular from one of the vertices of an equilateral triangle onto the opposite side We know that that perpendicular is going to bisect the opposite side. So hence we can say AD is equal to DC is equal to a by 2 Right, so perpendicular dropped from a vertex onto the opposite side bisects the opposite side in an equilateral triangle So AD is equal to this length is nothing but a by 2 So if this is a by 2 and the hypotenuse is a so we can find out length of BD by Pythagoras theorem So BD square is nothing but AB square minus AD square why because in a Pythagoras by Pythagoras theorem in a right-angle triangle Hypotenuse square is B square plus P square So hence we can find out P square from So we can find out P is nothing but h square minus B square. That's what I have used here So hence let us calculate the value of BD. Let us calculate the value of BD So BD is nothing but under root AB square, which is a square minus AD. AD is a by 2 whole square Which is then nothing but a square minus a square minus a square by 4 which is equal to root of 3a square by 4 which can be written as root 3a by 2 Correct now. Let us find out the value of sine 60 degrees. So sine of 60 degrees is nothing but opposite to opposite by hypotenuse. So hence it is AD upon AB correct AD I'm sorry not AD. It is it is BD BD BD here. So it will be It will be nothing but BD upon AB BD upon AB. So what is BD? We just found out it is nothing but root 3 by 2 A and what is AB? AB is A. So hence sine 60 is nothing but root 3 by 2 as A and A gets cancelled similarly cos of 60 degrees can be written as AD upon AB right cos of 60 is nothing but Adjacent that is AD upon Hypotenuse that is AB. So hence AD is nothing but A by 2 and divided by hypotenuse, which is A So hence it is 1 upon 2 similarly tan 60 now tan 60 degree is opposite by Adjacent so opposite opposite is BD and adjacent is AD. So hence it is BD upon AD Right. So BD is root 3 by 2 A and AD is A by 2 So hence it is nothing but root 3 Now the others are very easy. So cosecant 60 cosecant 60 is 1 upon sine. So it is 2 upon root 3 isn't it and cosecant 60 is equal to 1 upon cos cosine 60 or 1 upon cos 60 So hence it is 1 upon 1 by 2 which is 2 and cot 60 is 1 upon tan 1 upon tan 60 Which is 1 upon root 3 Correct. So hence these are the 6 t ratios for theta equals to 60 degrees Okay in the same Triangle we can also find out The values for 30 degrees. So let us see how so if this is 60 degrees and angle B is 60 also So hence when BD is a perpendicular dropped from B on to AC And BD bisects AC as well as angle ABC in a equilateral triangle. So this angle is 30 degrees Right. So now we can we can Write the t ratios again. So we had by Pythagoras theorem. We had figured out that A BD was equal to root 3 by 2 a so I'm writing it here root 3 by 2 a Right and again. So in and let us say AB is equal to a and AD is equal to a upon 2 Okay, now we have to find out sine 30 degrees. So if you see sine 30 degrees Is nothing but opposite to 30 is our ad side right ad here So it is a by 2 and divided by hypotenuse is a so sine 30 is half Correct. Similarly cos of 30 degrees is equal to adjacent. So adjacent to 30 is This side root 3 by 2 a so hence I will write root 3 by 2 a divided by Root 3 by 2 a divided by the hypotenuse which is a So hence cos 30 is root 3 by 2 correct now tan 30 is Opposite by adjacent. So opposite is a by 2 and adjacent is root 3 by 2 a Right. So hence it is 1 upon root 3 Right. Now others are very easy again. So if you find out cos he can 30 Is nothing but 1 upon sine 30 1 upon sine 30 degrees. So hence it is 2 right similarly secant 30 Is equal to 1 upon cos 30 So hence it is 2 upon root 3 And tan third sorry cot of 30 degrees is nothing but 1 upon tan 30 So hence it is root 3 So this is how we found out the trigonometric ratios of 30 degrees as well as 60 degrees So this table again you have to remember very important result. Similarly 60 for all 60 degrees Also, you have to remember this in the next session. We'll tell you how to Remember these numbers