subject: Reciprocal Trigonometric Ratio [print this page] In This article, we are going to learn about the definition of "trigonometry","what are trigonometric ratios?"
"what are reciprocal trigonometric ratios?""Examples for finding trigonometric ratios and reciprocal trigonometric ratios" one by one.
Definition of trigonometry:
The word "Trigonometry" is derived from two Greek words, Trigonon which is meant by "a triangle" and metron which is meant by "measure". So, the word Trigonometry means that "measurement of angles". The mathematical definition of "Trigonometry" is as follows.
Trigonometry is a branch of mathematics which deals with the relation between the angles and sides of a triangle.
For the figure shown in the example 1 find all the reciprocal trigonometric ratios.
Solution:
Cosec x = hypotenuse/ opposite side = BC / AC = 5 / 4
Sec x = hypotenuse / adjacent side = BC / AB = 5 / 3
Cot x = adjacent side / opposite side = AB / AC = 3 / 4.
The following points are very important for learning reciprocals,
Reciprocal is also known as the inverse some times.
Suppose we have to find the reciprocal of a number x, we can find the reciprocal is 1/x.
The resultant product of the two (that is the number and its reciprocal)is always 1. The reciprocal of the number is also known as the multiplicative inverse.
Reciprocal of 2 is 1/2 and reciprocal of x is 1/x. In the same way, reciprocal of 1/2 is 2 and that of 2/3 is 3/2.
To get the reciprocal of a real number, just divide 1 by it.
In this article we are going to learn about reciprocals and some example problems.
Learning some Basic Concepts in Reciprocals
The reciprocal of the imaginary number i (that is, + ) is -i. Reciprocal of -i(that is, - ) is i. This i times -i will produce 1. The multiplicative inverse of i and -i are the same as its additive inverse (which when added will give zero.
When you want to find the reciprocal of a mixed number like 13 1/2, first convert it into a regular fraction. Here it is 13+1/2 which is 27/2. Now interchange the numerator and denominator. Thus, the reciprocal of 27/2 is 2/27.
Learning some Example Problems in Reciprocals
Example 1:
Solve for x:
4/x2 =16
Taking reciprocal on both sides:
x2/4 =1/16
That is x2=4/16
Now, taking square root on both sides,
x=2/4(positive sqrt)
Divided by 2 we get
Answer x = 1/2
Example 2:
Solve 6/x = 5/3
Here, taking reciprocals on both sides,
x/6 = 3/5
x=3 * 6/5
Answer x=18/5
Example 3:
Solve for x: 1/x = 4
Here, just we have to take the reciprocal of 1/x, to get x
That is, x= 1/4
So, x=1/4
Answer x = 4
Example of Reciprocal Identities:
The reciprocal of -5 is -1/5, since -5(-1/5) = 1.
The opposite of 1/7 is -1/7, since 1/7 + (-1/7) = 0.
The reciprocal of 1/a is a, since a(1/a) = 1.
The opposite of -9 is 9, since -9 + 9 = 0.
The reciprocal of -b is -1/b, since -b(-1/b) = 1.
When you take the reciprocal, then the sign of the original number stays intact.
