Properties of Inverse Trigonometric Functions

The sine, cosine, tangent, cotangent, secant, and cosecant functions make up the fundamental building blocks of trigonometry. Inverse trigonometric functions are simply defined as the inverse functions of these fundamental building blocks. There are a few other names for them, including cyclometric functions, anti trigonometric functions, and arcus functions. When working with any of the trigonometry ratios, these inverse functions in trigonometry are what you need to employ to get the angle. The fields of engineering, physics, geometry, and navigation all make extensive use of the inverse trigonometric functions.

Inverse Trigonometric Functions

Inverse trigonometric functions are also referred to as “Arc Functions.” This is due to the fact that these functions, when applied to a specific value of trigonometric functions, create the length of arc that is required to obtain that value. The sine, cosine, tangent, cosecant, secant, and cotangent are examples of standard trigonometric functions. Inverse trigonometric functions do the opposite operation of these standard trigonometric functions. It is well known that the right triangle presents unique challenges for the solution of trigonometric functions. When the lengths of two of the triangle’s sides are already known, you can apply these six essential functions to determine the angle measure in the right triangle.

Inverse Trigonometric Formulas

The following are the formulas that have been combined together to make up the list of inverse trigonometric formulas. These formulas are important for converting one function to another, determining the principal angle values of the functions, and performing a wide variety of arithmetic operations over these inverse trigonometric functions. In addition, all of the fundamental trigonometric function formulae have been changed into the inverse trigonometric function formulas, and these formulas have been organised into the following four sets of formulas for your convenience.

• Arbitrary Values

• Reciprocal and Complementary functions

• Sum and difference of functions

• Double and triple of a function

Inverse Trigonometric Function Formulas for Arbitrary Values

The formula for the inverse trigonometric function for random values can be applied to any one of the six different trigonometric functions. When it comes to the inverse trigonometric functions of sine, tangent, and cosecant, the negatives of the values are interpreted as the function’s negative values. In addition, the negative values of the domain are expressed as the difference between the value of the function and the value of the cosine, secant, or cotangent function.

• sin^-1(-x) = -sin^-1x,x ∈ [-1,1]

• tan^-1(-x) = -tan^-1x, x ∈ R 

• cosec^-1(-x) = -cosec^-1x, x ∈ R – (-1,1)

• cos^-1(-x) = π – cos^-1x, x ∈ [-1,1]

• sec^-1(-x) = π – sec^-1x, x ∈ R – (-1,1)

• cot^-1(-x) = π – cot^-1x, x ∈ R

Inverse Trigonometric Function Formulas for Reciprocal Functions

The supplied inverse trigonometric function can be converted into its reciprocal function by using an inverse trigonometric function that works for reciprocal values of x. Because sin and cosecant are reciprocal to each other, tangent and cotangent are reciprocal to each other, and cos and secant are reciprocal to each other, this follows from the trigonometric functions.

Inverse sine, inverse cosine, and inverse tangent are the three terms that make up the inverse trigonometric formula. This formula can also be stated in the following forms:

• sin^-1x = cosec^-11/x, x ∈ R – (-1,1)

• cos^-1x = sec^-11/x, x ∈ R – (-1,1)

• tan^-1x = cot^-11/x, x > 0

• tan^-1x = – π + cot^-1 x, x < 0

Inverse Trigonometric Function Formulas for Complementary Functions

A right angle is an outcome when the complementary inverse trigonometric functions are added together. The sum of complementary inverse trigonometric functions is equal to a right angle when those functions are applied to the same values of x. Therefore, the sum of the sine and cosine functions, the tangent and cotangent functions, and the secant and cosecant functions is π/2. The interpretations of the complementary functions sine-cosine, tangent-cotangent, and secant-cosecant can be stated as follows:

• sin^-1x + cos^-1x = π/2, x ∈ [-1,1]

• tan^-1x + cot^-1x = π/2, x ∈ R

• sec^-1x + cosec^-1x = π/2, x ∈ R – [-1,1]

Sum and Difference of Inverse Trigonometric Function Formulas

According to the set of formulas that are provided below, it is possible to create a single inverse function by combining the sum and the difference of two inverse trigonometric functions. The formulas for the trigonometric functions sin(A + B), cos(A + B), and tan(A + B) can be used to obtain the sum and difference of the inverse trigonometric functions. These formulas for inverse trigonometric functions can be utilised in order to obtain the formulas for double and triple functions in a subsequent step.

• sin^-1x + sin^-1y = sin^-1(x.√(1 – y²) + y√(1 – x²))

• sin^-1x – sin^-1y = sin^-1(x.√(1 – y²) – y√(1 – x²))

• cos^-1x + cos^-1y = cos^-1(xy – √ (1 – x²).√(1 – y²))

• cos^-1x – cos^-1y = cos^-1(xy + √ (1 – x²).√(1 – y²))

• tan^-1x + tan^-1y = tan^-1(x + y)/(1 – xy), if xy < 1

• tan^-1x + tan^-1y = tan^-1(x – y)/(1 + xy), if xy > – 1

Double of Inverse Trigonometric Function Formulas

Following is a collection of formulas that can be used to solve the double of an inverse trigonometric function, which will result in the formation of a single trigonometric function.

• 2sin^-1x = sin^-1(2x.√(1 – x²))

• 2cos^-1x = cos^-1 (2x² – 1)

• 2tan^-1x = tan^-1(2x/(1 – x²))

Triple of Inverse Trigonometric Function Formulas

Following is a series of formulas that can be used to solve the triple of inverse trigonometric functions, which will result in the formation of a single inverse trigonometric function.

3sin^-1x = sin^-1 (3x – 4x³)

3cos^-1x = cos^-1 (4x³ – 3x)

3tan^-1x = tan^-1(3x – x³/1 – 3x²)

Tips and Tricks on Inverse Trigonometric Functions

There are many different formulas for inverse trigonometric functions, and some of the following pointers could be helpful in addressing these problems and using the formulas.

• sin^-1(sin x) = x, when -1 ≤ x ≤ 1

• sin(sin^-1x) = x, when -π/2 ≤ x ≤ π/2.

• sin^-1x is different from (sin x)^-1. Also (sin x)^-1 = 1/sinx

Conclusion

When at least two of the sides of a right triangle are known, the inverse trigonometric functions can be used to compute the measure of an angle in the triangle. The specific function that ought to be used is determined by the information that is known about the two sides. The inverse trigonometric functions perform the same function as the inverse trigonometric relations do; however, when an inverse function is utilised, because of its restricted range, it only offers one output for each input, and that output corresponds to whatever angle fits inside its range.