Angle is introduced and there by we can make a triangle.How trignometric ratios are intoduced?I am unable to understand myself why sinA=opp/hyp(rt anld trile).It is difficult to me to understand trignometric ratio though i saw hds of problems up to now.(sorry if i have asked you a wrong qestion)
Well, whenever you draw a triangle this shape, you'll always find that the length of the shortest side is exactly one-half the length of the longest side. (Try it and see!) So there's something special about that fraction , as far as this triangle is concerned: it is the ratio of the smallest side to the longest.
Now we call (as you probably know) the longest side of a right-angled triangle the hypotenuse; and you'll see that the smallest side in our triangle is opposite to the angle. So, as far as this angle of is concerned, we call this side the opposite (that seems fair, doesn't it?).
And what we have just discovered is that, for an angle of in a right-angled triangle, the ratio of the opposite side to the hypotenuse is exactly . We call this ratio the sine of the angle, and we can write this as a formula:
Well, if we now draw a right-angled triangle a different shape, we shall have a different angle and a different ratio, but the formula that I've just given you is the one that will tell you (if you know the size of the angle) what that ratio is.
Trigonometry is a very big subject, but that should be enough to get you started. We can define cosine and tangent in a similar way, using the name of the third side of the right-angled triangle, which is the adjacent.
These definitions are:
For angles greater than , you need an extended definition for sine, cosine and tangent. You can study the diagram that pacman has posted if you wish, or, for angles up to , you can use the formulae:
This will mean, for instance, that
For a more mathematically complete reply, you need to re-define sine, cosine and tangent, using a diagram similar to pacman's, which uses co-ordinate geometry. It goes something like this:
Consider a circle centre , radius unit. A is the point where the circle cuts the positive -axis. A point P then starts at A and moves anticlockwise around the circle. The angle is the angle through which the radius OP has turned.
Then we can define and as the - and -coordinates of P, respectively. In other words, P is the point whose co-ordinates are .
This, then, provides a definition of sine and cosine for all possible angles - including negative ones, and angles more than .
The definition of the tangent of an angle is then simply:
(You need to refine this a little to take account of what happens when , but that's another story!)