In Maths, we come across many different concepts such as arithmetic operations, geometry, algebra, calculus, trigonometry, statistics and probability, etc. Each concept is very important and gives us immense knowledge about using logic in real life. With the help of these concepts, we can easily do mathematical calculations quickly. Learning mathematics improves problem-solving skills.
All the topics in Maths are diverse and help to calculate different types of questions. But there are few topics which are interlinked with each other. For example, addition and subtraction both are different operations but they are related to each other. If we add 5 to 5 we get 10, 5 + 5 =10. But if we subtract 5 from 10 we get 5 again. So we can say, addition is the inverse function of addition.
Similarly, multiplication is the inverse function of the division. If we multiply a by b we get the result as ab, but if we divide ab by b then we get back ‘a’ as a result. One should not get confuse multiplicative inverse with the inverse of multiplication. Let us understand with an example.
Suppose we have to multiply 3 by 4, then we get 3 x 4 = 12. Now, the multiplicative inverse of 12 will give the value which when multiplied by 12 again, it will be equal to the identity. Hence, the multiplicative inverse of 12 is 1/12 or 12-1, such as;
12 x 1/12 = 1
Now if we have to find the inverse of multiplication of 3 x 4 = 12, then we need to divide 12 by 4 to get back 3 as an answer, such as;
12/4 = 3
In the same way, there are many inverse functions in Maths which returns the original value for which function gave the output. Let us say, if f(x) is a function then its inverse function is given by f-1(x). Let us take an example of algebra. Say f(x) = 2x+3 = y, where x and y is an arbitrary constant.
2x+3 = y
So we can write this function now as f-1(y) = (y-3)/2
In trigonometry, we have inverse functions such as inverse sine, inverse cosine and inverse tangent. But by inverse sine, we do not mean that it will be equal to its reciprocal, which is a cosine function. The inverse of any trigonometry function will return the value of the angle for which the trig function has generated the output. For example, sin 90 = 1 and sin-1 (1) = sin-1 (sin 90) = 90 degrees. Similarly, for inverse cos and inverse tan functions, the same definition applies. See examples below for better understanding.
Cos 90 = 0 ←→ cos-1(0) = cos-1 (cos 90) = 90 degrees
Tan 45 = 1 ←→ tan-1(1) = tan-1 (tan 45) = 45 degrees
But reciprocal functions are different from inverse functions. Let us see the reciprocals of trigonometric functions.
1/Sin A = Cosec A or 1/Cosec A = Sin A
So, if we multiply sine and cosec of the same angle, then the value will be equal to 1.
Sin A x Cosec A = Sin A x 1/Sin A = 1
Similarly, the reciprocals of cos and tan can be written as:
1/Cos A = Sec A
1/Tan A = Cot A
So, basically reciprocals are nothing but the multiplicative inverse of the value or function. Both are the same. These were some of the relationships between the mathematical functions.