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More about Inverse Trigonometric Ratios

More about Inverse Trigonometric Ratios

In this section, let's discuss about the allowed inputs and possible outputs for inverse trigonomteric ratios.

Allowed Inputs:

enlightenedThink: What will be the value of sin12? Is there any angle whose sine is 2?

The answer is NO! Since there is no angle whose sine is 2, it is absurd to talk about sin12. We can only apply the sin1 operation to those numbers which can actually be generated by the sine function.

For example, the following expressions are meaningless:

sin1(3),sin1(7)sin1(π),sin1(100)

The following expressions are all well-defined:

sin1(25),sin1(713)sin1(1719),sin1(23)

To summarize, the sin1 operation can be applied to only those numbers which fall in the interval [1,1]. The same holds true for the cos1 operation. However, since the tan function generates output over the set of all real numbers, the tan1 operation can be applied to any real number x, since there will always be an angle θ whose tan is x.

Using a similar line of reasoning, we conclude that:

  • The cot1 operation can be applied to any real number.

  • The sec1 and cosec1 operation can only be applied to numbers in the range (,1][1,), since the sec and cosec functions produce outputs in this range only.


Possible Outputs:

Let us analyze the characteristics of the outputs generated by the various inverse trigonometric ratio operations.

As an example, observe that

sinπ6=12,sin5π6=12

What value should we then assign to sin1(12)? Isn’t it correct to say that sin1(12)=π6 and also that sin1(12)=5π6? In fact, since there are an infinite number of angles whose sine is 12, the sin1 operation applied to 12 should seemingly give infinitely many values.

Similar remarks exist for the other inverse trigonometric operations. For example,

  • cos1(1) should give us all those angles whose cos is 1.

  • tan1(3) should give us all those angles whose tan is 3.

  • …and so on

However, for a number of reasons, we restrict the output of the inverse trigonometric operations in a way so that a unique output is generated in each case. For example, we will always write sin1(12)=π6 and not 5π6.

How do we decide which value to pick in any such situation? There is a principal range for each inverse trigonometric operation, and the output must lie in that principal range.

The principal ranges of the various inverse trigonometric operations are shown in the table below:

sin1

[π2,π2]

cos1

[0,π]

tan1

(π2,π2)

cot1

(0,π)

sec1

[0,π]{π2}

cosec1

[π2,π2]{0}

Note: The reason behind selecting these particular intervals for the principal range will become clear at a later stage.


Solved Examples:

Example 1: Which of the following are not well-defined mathematically? (a) sin1(17) (b) cos1(5) (c) tan1(11000) (d) cot1(99) (e) sec1(13) (f) cosec1(11).

Solution: The terms in (a), (c), (d) and (f) are well-defined. The term in (b) is not well defined since the argument is greater than 1. The term in (e) is not well-defined since the magnitude of the argument is less than 1.


Example 2: Identify the incorrect values:

(a) sin1(12)=3π4

(b) cos1(32)=π6

(c) tan1(13)=7π6

Solution: The values in (a) and (c) are incorrect, because they lie outside the principal range. The correct values should have been:

sin1(12)=π4,tan1(13)=π6


yesChallenge: Find the correct values of

(a) sin1(12)

(b) cos1(12)

(c) tan1(1)

Tip: The values should lie in the principal range of respective trigonomteric ratios.


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