Class 12th Inverse Trigonometric Functions Complete Formula Sheet

 

Class 12th Inverse Trigonometric Functions (ITF)

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# Inverse Trigonometric Functions (ITF) #

Inverse Trigonometric Functions (ITF), also known as Arc Functions, are the inverse operations of the basic trigonometric functions such as sin, cos, and tan. These functions are used to determine the angle when the value of a trigonometric ratio is known. In mathematics, they play a very important role in trigonometry, calculus, geometry, and physics.

# What are Inverse Trigonometric Functions?

• If a trigonometric function gives a ratio for a given angle, then an inverse trigonometric function gives the angle for a given ratio.

For example:

If sin θ = x, then θ = sin⁻¹(x)

If cos θ = x, then θ = cos⁻¹(x)

If tan θ = x, then θ = tan⁻¹(x)

• Here, the symbol ⁻¹ represents the inverse function (not the reciprocal).

# Main Inverse Trigonometric Functions #

• The six inverse trigonometric functions are:

1. sin⁻¹(x) – Inverse Sine (arcsin)

2. cos⁻¹(x) – Inverse Cosine (arccos)

3. tan⁻¹(x) – Inverse Tangent (arctan)

4. cot⁻¹(x) – Inverse Cotangent

5. sec⁻¹(x) – Inverse Secant

6. cosec⁻¹(x) – Inverse Cosecant

• Principal Value Branch :- 

Since trigonometric functions are periodic, they are not one-to-one over their entire domain. Therefore, we restrict their domains to define their inverses. The angle obtained from an inverse trigonometric function is called the principal value.

Examples:

Range of sin⁻¹(x) is [-π/2, π/2]

Range of cos⁻¹(x) is [0, π]

Range of tan⁻¹(x) is (-π/2, π/2)

# Applications of Inverse Trigonometric Functions #

• Inverse Trigonometric Functions are widely used in many fields such as:

Mathematics – solving trigonometric equations and identities

Calculus – differentiation and integration involving trigonometric forms

Physics – calculating angles in wave motion, optics, and mechanics

Engineering – used in signal processing, navigation, and structural analysis

Computer Graphics – calculating rotations and angles in 2D and 3D systems

• Conclusion :-

Inverse Trigonometric Functions are an essential part of higher mathematics. They help us determine angles from known trigonometric ratios and are widely used in theoretical and applied mathematics. Understanding ITF is important for students studying Class 11, Class 12 mathematics, engineering entrance exams, and advanced calculus.

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