Inverse Trigonometric Functions

What Are Inverse Trigonometric Functions?

Inverse trigonometric functions are the inverse of normal trigonometric functions. Alternatively denoted as cyclometric or arcus functions, these inverse trigonometric functions exist to counter the basic trigonometric functions, such as sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (cosec). When trigonometric ratios are calculated, the angular values can be calculated with the help of the inverse trigonometric functions.

How Do Inverse Trigonometric Functions Work?

The term Arcus functions, or Arc functions, is also used to denote inverse trigonometric functions. If a normal trigonometric function is being considered, it has a value. Using the inverse trigonometric function, we can calculate the arc length that is used to get that specific value. Whatever operation the basic trigonometric function performs, the inverse trigonometric function does exactly the opposite.

When considering right-angled triangles, the concept of trigonometry comes into play. Using the trigonometric functions, students can measure the angles created in the triangle by the base, height, or hypotenuse. Using the inverse trigonometric functions, the exact value of the created angle can be measured.

How Many Types of Inverse Trigonometric Functions Are There?

As stated before, the inverse trigonometric functions are the exact opposites of the basic trig functions. There are six basic functions in trigonometry. Every trigonometric ratio can be expressed with the help of these functions. As a result, there are also six inverse trigonometric functions, each acting as an inverse for the six trigonometric functions.

The inverse trigonometric functions are as follows-

Arcsine
Arccosine
Arctangent
Arccotangent
Arcsecant
Arccosecant

All of these six inverse trigonometric functions have been discussed in detail below, along with their ranges and domains.

What is Arcsine Function?

The arcsine function, or arcsin, is the first of the six inverse trigonometric functions. It is the inverse function that corresponds to the sine function. As a result, it is denoted by sin^-1 x. The arcsine function has a range that starts from -π/2 to π/2, and its domain lies from -1 to 1.

What is Arccosine Function?

The arccosine function, or arcos, is the inverse trigonometric function corresponding to the cosine or cos function in trigonometry. Hence, it is also denoted as cos^-1 x. The range of the arccosine function lies from 0 to π, and its domain starts at -1 and ends at 1.

What is Arctangent Function?

The arctangent function, or arctan, is the inverse trigonometric function corresponding to the tangent or tan function in trigonometry. In other words, it is the inverse of the tangent trig function. Therefore, it can also be denoted as tan^-1 x. Its range lies between -π/2 and π/2, and its domain lies between negative infinity (-∞) and positive infinity (∞).

What is Arccotangent Function?

The arccotangent function, or arccot, is the inverse of the cotangent or cot function. It can be represented as cot^-1 x. The range of the arccotangent function lies between 0 and π, and its domain lies between negative infinity and positive infinity.

What is Arcsecant Function?

The arcsecant function, or arcsec, is the inverse of the secant or sec function. Hence, it can be represented as sec^-1x. The range of the arcsecant function lies between -π/2 and π/2, excluding zero. The domain either lies from negative infinity to -1, or from 1 to positive infinity. The values near and at zero are excluded.

What is the Arccosecant Function?

The arccosecant function, or arccos, acts as the inverse of the cosecant or the cosec function. As a result, it can be denoted as cosec^-1 x. The domain and range of the arccosecant function are the same as the arcsecant function. Hence, its range lies between -π/2 and π/2, excluding zero, and its domain lies either from negative infinity to -1, or from 1 to positive infinity. Here also, the values at and close to zero are excluded.

Formulas

All of the inverse trigonometric functions that exist come with their own set of formulae. Students need to be familiar with these formulas to solve complex inverse trigonometric problems quickly.

Some of the most basic inverse trigonometric function formulas are given below.

Arcsine Function

For any arcsine function x,

{sin}^{- 1} (- x) = - {sin}^{- 1} (x)

Here, x lies from -1 to 1.

Arccosine Function

For any arcos function x,

\cos^{- 1} (- x) = π - {cos}^{- 1} (x)

Here also, x lies from -1 to 1.

Arctangent Function

For any arctan function x,

{tan}^{- 1} (- x) = - {tan}^{- 1} (x)

Here, x lies in the real number set R.

Arccotangent Function

For any arc cot function x,

\cot^{- 1} (- x) = π - {cot}^{- 1} (x)

Here also, x lies in the real number set R.

Arcsecant Function

For any arcsec function x,

\sec^{- 1} (- x) = π - {sec}^{- 1} (x)

Here, the mod x, or |x| is always greater than or equal to 1.

Arccosecant Function

For any arccosec function x,

c o s e c^{- 1} (- x) = - {cosec}^{- 1} (x)

Here also, |x| is always greater than or equal to 1.

These are some of the basic inverse trig function formulas that come in very handy during operations.

What Are the Derivatives of Inverse Trigonometric Functions?

Just like normal trig functions, inverse trigonometric functions can also be differentiated. By differentiation, the first-order derivatives of the inverse trigonometric functions can be found.

All of the six inverse trigonometric functions have their first-order derivatives. They are given below.

Arcsine Function

For y = sin^-1 x,

\frac{d y}{d x} = \frac{1}{\sqrt{1 - x^{2}}}

Arccosine Function

For y = cos^-1 x,

\frac{d y}{d x} = \frac{- 1}{\sqrt{1 - x^{2}}}

Arctangent Function

For y = tan^-1 x,

\frac{d y}{d x} = \frac{1}{1 + x^{2}}

Arccotangent Function

For y = cot^-1 x,

\frac{d y}{d x} = \frac{- 1}{1 + x^{2}}

Arcsecant Function

For y = sec^-1 x,

\frac{d y}{d x} = \frac{1}{[|x| \sqrt{(x^{2} - 1)}]}

Arccosecant Function

For y = cosec^-1 x,

\frac{d y}{d x} = \frac{- 1}{[|x| \sqrt{(x^{2} - 1)}]}

Practice Problem

Find the value of sin (cos^-1 4/5)

⇒ For assumption, let:

\begin{array}{l} {⇒cos}^{- 1} \frac{4}{5} = x \\ = > ∴ cos x = \frac{4}{5} \\ = > {Using the formula sin}^{2} {x + cos}^{2} x = 1, \\ =>sin x = \sqrt{1 - {cos}^{2} x} \\ = > ∴ sin x = \sqrt{1 - {(\frac{4}{5})}^{2}} \\ = > sin x = \sqrt{1 - (\frac{16}{25})} \\ = > sin x = \sqrt{\frac{25 - 16}{25}} \\ = > sin x = \sqrt{\frac{9}{25}} \\ = > sin x = \frac{3}{5} \end{array}

Therefore, sin (cos^-1 4/5) = 3/5.

Context and Applications

Inverse trigonometric functions have some major real-life applications. Examples include operations in navigating, processes in geometry, describing terms in physics, and applications in engineering work.

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for:

Bachelors of Science in Mathematics
Masters of Science in Mathematics

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