also of other trigonometric ratios) remains the same in QAN also. From our observations, it is now clear that the values of the trigonometric ratios of an angle do not vary with the lengths of the sides of the triangle, if the angle remains the same. Note : For the sake of convenience, we may write sin A, cos A, etc., in place of (sin A) , (cos A) , etc., respectively. But cosec A = (sin A) – sin – A (it is called sine inverse A).
sin – A has a different meaning, which will be discussed in higher classes. Similar conventions hold for the other trigonometric ratios as well. Sometimes, the Greek letter (theta) is also used to denote an angle. We have defined six trigonometric ratios of an acute angle.
If we know any one of the ratios, can we obtain the other ratios? Let us see. If in a right triangle ABC, sin A = , then this means that BC , i.e., the lengths of the sides BC and AC of the triangle ABC are in the ratio : (see Fig. .
). So if BC is equal to k , then AC will be k , where k is any positive number. To determine other trigonometric ratios for the angle A, we need to find the length of the third side AB. Do you remember the Pythagoras theorem?
Let us use it to determine the required length AB. AB = AC – BC = ( k ) – ( k ) = k = ( k ) Therefore, AB = k So, we get AB = k (Why is AB not – k ?) Now, cos A = AB Similarly, you can obtain the other trigonometric ratios of the angle A. Fig. .
Remark : Since the hypotenuse is the longest side in a right triangle, the value of sin A or cos