[Maths Class Notes] on Inverse Trigonometric Functions Pdf for Exam

We already know about inverse operations. As we know addition and subtraction are inverse operations, and multiplication and division are inverse operations. Each operation has the opposite of its inverse. Similarly, inverse functions of the basic trigonometric functions are said to be inverse trigonometric functions. It also termed as arcus functions, anti trigonometric functions or cyclometric functions. 

The inverse of g is denoted by ‘g -1’.

Let y = f(y) = sin x, then its inverse is y = sin-1x.

In this article let us study the inverse of trigonometric functions like sine, cosine, tangent, cotangent, secant, and cosecant functions. 

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Introduction to Inverse Trig Function

The inverse trigonometric functions actually perform the opposite operation of the trigonometric functions such as sine, cosine, tangent, cosecant, secant, and cotangent. We know that trig functions are especially applicable to the right angle triangle. These six important functions are used to find the angle measure in a right triangle when two sides of the triangle measure are known.

The convention symbol represents the inverse trigonometric function using arc-prefixes like arcsin(x), arccos(x), arctan(x), arccsc(x), arcsec(x), arccot(x). 

• sin-1x, cos-1x, tan-1x etc. denote angles or real numbers whose sine is x, cosine is x and tangent is x, provided that the answers given are numerically smallest available. These are also termed arcsin x, arccosine x etc.

• If there are two angles, one positive and the other negative having the same numerical value, then a positive angle should be taken.

• Principal values, domains of inverse circular functions and range of inverse trig functions:

Domain and Range

In the sine function, many different angles [theta] map to the same value of [sin(theta)]. For example,

0=sin0=sin(π)=sin(2π)=⋯=sin(kπ)

for any integer kk. To overcome the problem of having multiple values map to the same angle for the inverse sine function, we will restrict our domain before finding the inverse.

The graphs of the inverse functions are the original function in the domain specified above, which has been flipped about the line y=xy=x. The effect of flipping the graph about the line y=xy=x is to swap the roles of xx and yy, so this observation is true for the graph of an inverse function. See the wiki Inverse Trigonometric Graph for more details.

Inverse Trigonometric Functions Graphs

There are particularly six inverse trig functions for each trigonometric ratio. The inverse of six important trigonometric functions are:

• Arcsine

• Arccosine

• Arctangent

• Arccotangent

• Arcsecant

• Arccosecant

Graphs of all Inverse Circular Functions

1. Arcsine

 y = sin-1x, |x| ≤ 1, y ∈ −π/2,π/2

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• sin-1x is bounded in −π/2,π/2

• sin-1x is an increasing function.

• In its domain, sin-1x attains its maximum value π/2 at x = 1 while its minimum value is -π/2 which occurs at x = -1.

2. Arccosine

 y = cos-1x, |x| ≤ 1, y ∈ 0,π

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• cos-1x is bounded in 0,π

• cos-1x is a decreasing function.

• In its domain, cos-1x attains its maximum value π at x = -1 while its minimum value is 0 which occurs at x = 1.

3. Arctangent

 y = tan
-1x,where  x ∈ R, y ∈ (-[frac{pi}{2}][frac{pi}{2}])

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• tan-1x is bounded in (-π/2, π/2).

• tan-1x is an increasing function.

 

4. Arccotangent

 y = cot-1x where x ∈ R, y ∈ (0, π)

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• cot-1x is bounded in (0, π).

•  cot-1x is a decreasing function.

 

5. Arccosececant

 y = cosec-1x, |x| ≥ 1, y ∈ (−π/2,0)∪(0,π/2)

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• cosec-1x is bounded in −π/2,π/2

• cosec-1x is a decreasing function.

• In its domain, cosec-1x attains its maximum value π/2 at x = 1 while its minimum value is -π/2 which occurs at x = -1.

 

6. Arcsecant

y = sec-1x, |x| ≥ 1, y ∈ (0,π/2)∪(π/2,π)

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1. sec-1x is bounded in 0,π

2.  sec-1x is an increasing function.

3. In its domain,sec-1x attains its maximum value π at x = -1 while its minimum value is 0 which occurs at x = 1.

Here is the list of all the inverse trig functions with their notation, definition, domain and range of inverse trig functions.

Inverse Trigonometric Functions Table

The derivatives of inverse trig functions are first-order derivatives. Let us check here the derivatives of all the six inverse functions.

Inverse Trigonometric Functions Derivatives

Solved Examples

Example 1: Find the value of tan-1(tan 9π/ 8 )

Solution:

 tan-1(tan9π/8)       

= tan-1tan ( π + π/8)

 = tan-1 (tan(π/ 8))

=π/ 8

Example 2: Fi
nd sin (cos-13/5).

Solution:

Suppose that, cos-1 3/5 = x

So, cos x = 3/5

We know, sin x =  [sqrt{1 – cos2x}]

So, sin x = [sqrt{1 – 9/25}] = 4/5

This implies, sin x = sin (cos-1 3/5) = 4/5

Solved example

A tower, 28.4 feet high, must be secured with a guy wire anchored 5 feet from the base of the tower. What angle will the guy wire make with the ground?

Draw a picture.

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tanθtanθtanθtan−1(tanθ)θ=opp.adj.=28.45=5.68=tan−1(5.68)=80.02.

The following problem that involves functions and their inverses will be solved using the property f(f−1(x))=f−1(f(x)). In addition, technology will also be used to complete the solution.

Important Formulae :

1. sin-1 (1x) = cosec-1 x,   x ≥ 1 or x ≤ -1

2.  cos-1 (1x) = sec-1 x,   x ≥ 1 or x ≤ -1

3.  tan-1 (1x) = cot-1 x,   x ∈ R

4.  cos-1 (-x) = π – cos-1 (x),   x ∈ [ -1 , 1 ]

5.  sec-1 (-x) = π – sec-1 (x),   | x | ≤ 1

6.  cot-1 (-x) = π – cot-1 (x),   x  ∈ R

7.  sin-1 (-x) = – sin-1 (x),   x  ∈ [ -1 , 1]

8.  tan-1 (-x) = – tan-1 (x),   x  ∈ R

9.  cosec-1 (-x) = – cosec-1 (x),   | x |  ≥ 1

10.  sin-1 (x) + cos-1 (x) =   π2 x

11.  tan-1 (x) + cot-1 (x) =   π2 x

12.  cosec-1 (x) + sec-1 (x) =   π2 |x| ≥ 1

13.  tan-1 (x) + tan-1 (y) =  tan-1 (x+y1–xy) xy < 1

14.  tan-1 (x) – tan-1 (y) =  tan-1 (x–y1+xy) xy > -1

15.  tan-1 x = sin-1  (2×1+x2)

16.  tan-1 x = cos-1  (1–x21+x2)

17.  tan-1 x = tan-1  (2×1–x2)

18.  sin-1  (2x 1–x2−−−−√) = 2sin-1  x

19.  sin-1  (2x 1–x2−−−−√) = 2cos-1  x

20.  sin-1  x = sin-1  (3x – 4×3)

21.  cos-1  x = cos-1  (4×3 – 3x)