Thursday, August 28, 2014

Basics of Trigonometry - 09

Blog-90

Dear Students,

Now we will study the next part of trigonometry. 
Every triangle has three sides so there will be six ratios of the lengths of the sides of a triangle which are already known to us. These are the trigonometric ratios. In the adjacent diagram, we have the following ratios. (Let us take Angle A for all the trigonometric ratios).:

1) BC/AC = sin A
2) AB/AC = cos A
3) BC/AB = tan A
4) AC/BC = csc A
5) AC/AB = sec A
6) AB/BC = cot A

In a right angled triangle, by theorem of Pythagoras, we have, BC AB 2   = AC 2   [ Please see the adjacent diagram]
If we divide above equation by AC 2 , AB 2  and BC 2 , 
we get three different identities.
1) Divide equation (1) by AC 2 , we get 
BC AB 2   = AC 2      ----------  (1)

   (BC 2/AC 2) + (AB 2/AC 2)  = (AC 2/AC 2)
    (BC/AC)2 + (AB/AC)2   = (AC/AC)2
    (sin A)2 + (cos A)2   = (1)2
   sin 2 A + cos 2 A   = 1

   sin 2 A + cos 2 A   = 1 

2) Divide equation (1) by AB 2 , we get 
BC AB 2   = AC 2      ----------  (1)

   (BC 2/AB 2) + (AB 2/AB 2)  = (AC 2/AB 2)
    (BC/AB)2 + (AB/AB)2   = (AC/AB)2
    (tan A)2 + (1)2   = (sec A)2
   tan 2 A + 1   sec 2 A 

   1 + tan 2 A  sec 2 A
3) Divide equation (1) by BC 2 , we get 
BC AB 2  = AC 2      ----------  (1)

   (BC 2/BC 2) + (AB 2/BC 2)  = (AC 2/BC 2)
    (BC/BC)2 + (AB/BC)2  = (AC/BC)2
    (1)2 + (cot A)2 = (csc A)2
   1 + cot 2 A = csc 2 A 

   1 + cot 2 A csc 2 A

Dear students,
Here you will have to remember following things so that you will understand all the trigonometric proofs.

1) For the given right angled triangle, write the formula of Pythagoras theorem.
2) Divide both the sides of Pythagoras formula for the given right angled triangle by all the three sides one by one to get all the three trigonometric ratios.
3) Now you will get all the three trigonometric identities as under:
     a) sin 2 A + cos 2 A = 1
     b) 1 + tan 2 A = sec 2 A
     c) 1 + cot 2 A = csc 2 A 

So, Students, you will also find some fantastic way of presenting your studies to improve your thinking skills.

Anil Satpute