 Hello friends so In this video, we are going to discuss the law of science Most of you know that in a given triangle a by sine a equals b by sine b equals c by sine c and It is equal to twice the circum radius. So you Understand what circum radius means so if triangle ABC is inscribed in a circle Then r is the radius of that circle. Okay, so here we have taken two cases Where in the first case ABC is an acute angled triangle and in the second case ABC is a obtuse Angled triangle so in both the cases, let's see if you can prove it Through this method or through this method of geometry in your trigonometric classes. You would have proved it Using plain, you know application of trigonometric ratios and all so here. We'll try to use geometry as well To prove this so how to go about it. So if you take let's let's see Let's say this is case one or this is a figure one and this is figure two. Okay, so Let's understand how we go about it. So if you see in In Triangle ABC, right? I have denoted the sides as a b and c small a b and c you can see in the figure Now I have joined CD now CD happens to be a CD is the dia is the diameter in both the cases Diameter in both the cases and this is equal to to our is it twice the radius, right now clearly angle C B D CBB in both the cases is equal to 90 degree right angles Why because angle subtended by a diameter on any part of the circumference is 90 degree So CBD in both the cases is a 90 degree now in Triangle CBD CBD I can say Sign of B or other sign of Yeah, sign of B is Equal to sign of B or sorry not sign of B. So in triangle ABD, we can say sign of D rather So sign of angle D sign D will be how much you can see opposite by her partner. So a upon to our Okay, and clearly Clearly angle D is equal to angle a Is it angle D is equal to angle a why because In and this is in case one angle D is equal to angle a in case one why because angle subtended Angle subtended angle subtended by a chord Accord on the same segment on the same segment or Same segment are equal We know that right on the same, you know, same part circumference of the same segment now If you see clearly angle D is equal to angle a and and and hence hence in this case sign D It will be equal to sign a right and in case two if you see In case to ABCD is a cyclic quadrilateral is a cyclic Cyclic quadrilateral quadrilateral that means in cyclic quadrilateral We know that opposite angles are supplementary. So hence angle a plus angle D is 180 degrees All right, so hence Angle a or rather angle D is equal to 180 degrees minus angle a therefore sign D Will be equal to sign of 180 degrees minus angle a Correct, that means sign 180 minus theta is sin theta. So hence it is sine a so in both the cases I'm D is equal to sine a right and what was sine D basically a by 2 are you can see from here So hence in both the cases Sine a will be equal to a by 2r right this implies a upon sine a Is equal to 2r? Correct similarly, you can say similarly the other three ratio other two ratios rather similarly You can also say a by sine B. You will you can do the same exercise. You'll get a by 2b is 2r and Sorry, not a by 2b my bad not a by sine b. It's B by sine B B by sine B is 2r and similarly you will get C by sine C is Also equal to 2r Correct, so hence in a nutshell I can say in any given triangle a upon sine a will be equal to B upon sine B and This will be equal to C upon Sine C and all of them are equal to twice the Circum radius. So this is what the result is Okay, so you learned this, you know This proof using geometry properties of circle as well as the basic trigonometry right now there will be few questions which will be attached to The application of sign rule you can find them in the description and Try to solve them using the sign rule