Inverse Trigonometric Functions

Inverse trigonometric functions, or arc functions, are majorly applied in the fields of engineering, physics, geometry, etc. Trigonometry helps us understand the relationship between angles and sides of a triangle and is vastly used in navigation, graphics, physics, and astronomy. Some of the basic trigonometric functions are sin, cos, tan, cosec, sec, and cot. The inverse trigonometric functions, on the other hand, are denoted as sin⁻¹x, cos⁻¹x, cot⁻¹x, tan⁻¹x, cosec⁻¹x, and sec⁻¹x. Let’s understand in detail about these functions, their formulas, and their applications.

Table Of Contents

• What are inverse trigonometric functions?
• Inverse Trigonometric Formulas
• Inverse Trigonometric Functions Table
• Importance of Trigonometric Functions
• Examples
• (FAQs)
  
  

What are inverse trigonometric functions?

Inverse trigonometric functions, often called arc-functions or anti-trigonometric functions, allow us to determine an angle from a given trigonometric ratio (e.g., sine, cosine, or tangent). Since trigonometric functions repeat values, each inverse is defined with a restricted domain (called the principal value) to ensure they are single-valued. The six main ones are

arcsin (sin⁻¹) – domain: [−1,1], range: [−π/2, π/2]

arccos (cos⁻¹) – domain: [−1,1], range: [0, π]

arctan (tan⁻¹) – domain: ℝ, range: (−π/2, π/2)

arccot, arcsec, and arccsc, each with their own defined ranges.

Inverse Trigonometric Formulas

In this section, we have listed the categories in which the inverse trigonometric formulas are grouped under. These formulas are helpful to convert one function to another, to find the principal angle values of the functions, and to perform numerous arithmetic operations across these inverse trigonometric functions. Further, all the basic trigonometric function formulas have been transformed to the inverse trigonometric function formulas and are classified here as the following four sets of formulas.

1. Arbitrary Values

2. Reciprocal and Complementary functions

3. Sum and difference of functions

4. Double and triple of a function

Inverse Trigonometric Functions Table 

Importance of Inverse Trigonometric Functions

Inverse trigonometric functions are more than just mathematical entities—they allow you to model real-world systems with precision. Along with being a key tool across geometry, calculus, engineering, physics, and computer science, familiarity with inverse trigonometric functions can become a powerful asset in your analytical toolkit.

1. Solving for Angles in Right Triangles

If you know two sides of a right triangle, inverse trigonometric functions let you find the angle. For instance, θ=arctan(adjacent/opposite) is used in engineering when calculating slopes, roof angles, or navigation.

2. Advanced Mathematics

Inverse trigonometric functions appear in calculus (e.g., derivatives, integrals), especially in integrals. 

3. Applications in Physics, Engineering & Navigation

They’re essential in signal processing, robotics, satellite tracking, and map navigation: converting between coordinate systems and determining orientation angles.

4. Computer Science & Algorithms

Used in graphics, game development, and data modeling whenever converting a ratio back to an angle.

Solved Sample Problems for Inverse Trigonometric Functions

Problem 1: Find the principal value of cos⁻¹(−1/2).

Solution:

Let us assume that x = cos⁻¹(−1/2).

We can write this as

cos x =  −1/2

The range of the principal value of the inverse trig function cos⁻¹ is [0, π].

Thus, the principal value of cos⁻¹(−1/2) is 2π/3.

Answer: Hence, the principal value of cos⁻¹(−1/2) is 2π/3.

Problem 2: Find the value of x for sin(x) = 2.

Solution: Given: sin x = 2

x = sin⁻¹(2), which is not possible.

Hence, there is no value of x for which sin x = 2, so the domain of sin⁻¹x is −1 to 1 for the values of x.

Frequently Asked Questions

1: What is the difference between sin⁻¹(x) and 1/sin(x)?
Anwser: They are entirely different. Sin⁻¹(x) = arcsin(x), the inverse function. 1/sin(x) = cosecant, csc(x).

2: Why do inverse trig functions have restricted ranges?
Anwser: Because sine, cosine, and tangent are periodic and not one-to-one. To make a valid “undo” function, we limit the input range so each inverse is single-valued.

3: Can arcsin(2) or arccos(3) be calculated?
Anwser: No. Since sine and cosine never exceed 1 in magnitude, their inverses are undefined outside ±1.

4: How do I evaluate arccos(−√3/2)?
Anwser: It equals 5π/6(150°) because the arccos range is [0, π].