Sine inverse trigonometric function is given by :

The composition

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Trigonometric and inverse trigonometric functions are inverse to each other. We can use them to compose new functions. In such composition, trigonometric function represents value of trigonometric ratio, whereas inverse trigonometric function represents angle. The composite function either evaluates to value or angle, depending on particular composition.

Sine inverse trigonometric function is given by :

The composition

We shall discuss this composition with respect to individual inverse trigonometric ratio.

Sine inverse trigonometric function is given by :

In order to maintain generality, we replace y by x as :

The composition

However, x as argument of sine function can assume angle values belonging to real number set. It means angles represented by LHS and RHS can be different if we consider angle values beyond principal set selected to render corresponding trigonometric function invertible.

sine function |
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Let us consider adjacent intervals such that all sine values are included once. Such intervals are

sine function |
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Our task now is to determine angles in any of these new intervals, say

Value diagrams |
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Hence,

In order to find expression corresponding to negative angle interval

Hence,

Combining three results,

```
|-π-x; x∈ [-3π/2, -π/2]
sin⁻¹ sinx = | x; x∈ [-π/2, π/2]
| π- x; x∈ [π/2, 3π/2]
```

We can similarly find expressions for more such intervals.

Using three expressions obtained above, we can draw plot of the composition function. We extend the plot, using the fact that composition is a periodic function with a period of 2π. The equation of plot, which is equivalent to plot y=x shifted by 2π towards right, is :

The equation of plot, which is equivalent to plot y=x shifted by 2π towards left, is :

sine inverse of sine |
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We see that graph of composition is continuous. Its domain is R. Its range is

The composition

Let us consider adjacent intervals such that all cosine values are included once. Such intervals are [π, 2π], [2π, 3π] etc on the right side and [-π, 0], [-2π, -π] etc on the left side of the principal interval.

cosine function |
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The new interval

Value diagrams |
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Hence,

In order to find expression corresponding to negative angle interval

Hence,

Combining three results,

```
|-x; x∈ [-π, 0]
cos⁻¹ cosx = | x; x∈[0, π]
|2π- x; x∈ [π, 2π]
```

We can similarly find expressions for other intervals.

Using three expressions obtained above, we can draw plot of the composition function. We have extended the plot, using the fact that composition is a periodic function with a period of 2π. The equation of plot, which is equivalent to plot y=x shifted by 2π towards right, is :

The equation of plot, which is equivalent to plot y=x shifted by 2π towards left, is :

cosine inverse of cosine |
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We see that graph of composition is continuous. Its domain is R. Its range is

The composition

Let us consider adjacent intervals such that all tangent values are included once. Such intervals are (π/2, 3π/2), (3π/2, 5π/2) etc on the right side and (-3π/2, -π/2), (-5π/2, -3π/2) etc on the left side of the principal interval.

tangent function |
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The new interval

Value diagrams |
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Hence,

In order to find expression corresponding to negative angle interval

Hence,

Combining three results,

```
| x+π; x∈ (-3π/2, -π/2)
tan⁻¹ tanx = | x; x∈ (-π/2, π/2)
| x-π; x∈ (π/2, 3π/2)
```

We can similarly find expressions for other intervals.

Using three expressions obtained above, we can draw plot of the composition function. We have extended the plot, using the fact that composition is a periodic function with a period of π. The equation of plot, which is equivalent to plot y=x shifted by π towards right is :

The equation of plot, which is equivalent to plot y=x shifted by π towards left is :

These results are same as obtained earlier. It means that nature of plot is same in the adjacent intervals.

tangent inverse of tangent |
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We see that graph of composition is discontinuous. Its domain is

The composition

Let us consider adjacent intervals such that all cosine values are included once. Such intervals are [π/2, 3π/2] – {π}, [3π/2, 5π/2] – {2π} etc on the right side and [-3π/2, -π/2] – {-π}, [-5π/2, -3π/2] – {-2π} etc on the left side of the principal interval.

cosecant function |
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The new interval

Value diagrams |
---|

Hence,

In order to find expression corresponding to negative angle interval

Hence,

Combining three results,

```
|- π-x; x∈[-3π/2, -π/2] – {-π}
cosec⁻¹ cosecx = | x; x∈[-/2, π/2]-{0}
| π- x; x∈[π/2, 3π/2] – {π}
```

We can similarly find expressions for other intervals.

Using three expressions obtained above, we can draw plot of the composition function. We have extended the plot, using the fact that composition is a periodic function with a period of 2π. The equation of plot, which is equivalent to plot y=x shifted by 2π towards right, is :

The equation of plot, which is equivalent to plot y=x shifted by 2π towards left, is :

cosecant inverse of cosecant |
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We see that graph of composition is discontinuous. Its domain is

We can similarly find out expressions for different intervals for arcsecant and arccotangent compositions. We have left out discussion of these two functions as exercise.

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