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This is the most important triangle because we get to know the trigonometric ratios from here. Tan x, Sin x & Cos x are ratios or fractions created out of the three sides of a right angled triangle. We are assuming that the students already know the definitions of the trigonometric ratios. 

To start with our first lesson look at this diagram

See that the hypotenuse is a and the side adjoining the angle x is aCos x and the side opposite to the angle x is aSin x. Please learn to use this theorem. This is extremely useful. We will see another version of this below. By using these you can find the two other sides of a right angled triangle immediately when you know the length of the hypotenuse. 
In the diagram above, we are using the other angle y. Thus, again following the same principle, the side adjoining the angle is aCos y and the side opposite the angle y is aSin y. Please understand this well. 

Next we want you to look at the diagram again and notice that this right angled triangle is standing upright. But if we rotate the triangle in whichever direction, the three sides: a, aSiny and aCosy remain the same. This can be called geometric vision and is very helpful. Look at the diagrams below. 

Now please see the Exercise below and state the two sides of the right angled triangles shown as fast as possible.

​Remember only two things: 1) if the side is next to the angle, use Cos and 2) if the side is opposite to the angle, use Sin.

That’s all you need to remember. And don’t forget to multiply with the hypotenuse. [We have used different alphabets for denoting the hypotenuse. Get used to that. Know that an alphabet is just a placeholder for a number. When you see a or k or t or m in these diagrams, assume that they are some number like 5, 7, 27, 33. Initially this will make it easier for you to understand. Later you will get very used to it.]

Go to Geometry Lesson 2