Thursday, September 26, 2013

Basics of Trigonometry - 02

Blog-83


Dear Students,

Trigonometry is very important topic for your higher studies. We must study this topic with all basics. Actually this topic is very simple. If we concentrate to learn this topic from beginning, we will definitely understand the basics of this topic and it will help us to remember the formulas (formulae) very easily. In the end of all the Blogs related to “Basics of trigonometry”, we will list of all the formulas (formulae) in the separate sheet so that it will be help for you to remember all these formulas (formulae).


In Blog-82, we asked the following two questions. We are sure that you might got the answers of these questions.  We will see both the answers one by one.

1) Define all trigonometric ratios for an angle C of above triangle ABC .  
Answer: 
1) sin C  = AB/AC
2) cos C = BC/AC
3) tan C  = AB/BC
4) csc C = AC/AB
5) sec C  = AC/BC
6) cot C  = BC/AB  


2) Observe the adjacent figure and write down all the trigonometric ratios.

Answer:
1) sin P   = RQ/PQ          1) sin Q   = PR/PQ
2) cos P  = PR/PQ           2) cos Q  = RQ/PQ 
3) tan P   = RQ/PR           3) tan Q  = PR/RQ
4) csc P  = PQ/RQ           4) csc Q  = PQ/PR
5) sec P  = PQ/PR            5) sec Q = PQ/RQ
6) cot P   = PR/RQ           6) cot Q  = RQ/PR  
In above example, just observe 
case 1) sin P and cos Q
case 2) cos P and sin Q

Here, basic point to understand is this that the comparison of two angles is given with different trigonometric ratios. First observe these two angles. angle P and angle Q. In a right angled triangle, PQR, Angle R = 90, so remaining two angles P and Q are complementary angles. 

Here all of us will remember following trigonometric relations of complimentary angles of a right angled triangle.

1) sin A  = cos (90-A)
2) cos A = sin (90-A)
3) sec A = csc (90-A)
4) csc A = sec (90-A)
5) tan A  = cot (90-A)
6) cot A  = tan (90-A)

A) Easy way to remember the above relations:

Relation between the trigonometric ratios of co-terminal angles. (See the following figure)



Trigonometric Ratios of an angle A
Trigonometric Ratios of Complimentary Angle of A. (90°-A)
Ratio (In the form of length of the sides of a triangle)
sin A
cos (90°-A)
BC/AC
cos A
sin (90°-A)
AB/AC
csc A
sec (90°-A)
AC/BC
sec A
csc (90°-A)
AC/AB
tan A
cot (90°-A)
BC/AB
cot A
tan (90°-A)
AB/BC


B) Relation between the trigonometric ratios:

Observe the following trigonometric ratios from the figure given bellow:




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



If we divide equation 1 by equation 2 we get equation 3. i.e. 
                            AB/AC
sin C / cos C = ------------
                            BC/AC

                            AB
sin C / cos C = -------
                            BC                

sin C / cos C = tan C

In the same way we can prove that 

cos C / sin C  =  cot C

In the Next Blog, we will study the trigonometric ratios of some standard angles and trigonometric identities.

Anil Satpute