Inverse Trigonometric Functions Class-12th Board

Inverse Trigonometric Functions Class-12th Board: Inverse trigonometric functions are an essential part of the Class 12 curriculum, and they play a significant role in higher-level mathematics, including calculus, geometry, and real-world applications. This chapter extends the concepts of trigonometric functions by exploring their inverses.

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Introduction to Inverse Trigonometric Functions Class-12th Board

Inverse trigonometric functions are the inverse operations of the trigonometric functions, such as sine, cosine, and tangent. These functions are used to determine the angle when the trigonometric ratio is known. The primary inverse trigonometric functions are:

• sin⁻¹(x) (also called arcsin(x)): Inverse of the sine function.
• cos⁻¹(x) (also called arccos(x)): Inverse of the cosine function.
• tan⁻¹(x) (also called arctan(x)): Inverse of the tangent function.
• cosec⁻¹(x) (also called arccosec(x)): Inverse of the cosecant function.
• sec⁻¹(x) (also called arcsec(x)): Inverse of the secant function.
• cot⁻¹(x) (also called arccot(x)): Inverse of the cotangent function.

Principal Value in Inverse Trigonometric Functions Class-12th Board

Inverse trigonometric functions return multiple values for a given trigonometric ratio. However, for the sake of consistency and well-defined operations, we restrict their domains and ranges to what is called the principal value branch.

Principal Values:

sin^-1 x for x ∈ [-1, 1] gives values in [-π/2, π/2]

cos^-1 x for x ∈ [-1, 1] gives values in [0, π]

tan^-1 x for x ∈ R gives values in (-π/2, π/2)

cosec^-1 x for x ∈ (-∞, -1] ∪ [1, ∞) gives values in [-π/2, π/2]

sec^-1 x for x ∈ (-∞, -1] ∪ [1, ∞) gives values in [0, π]

cot^-1 x for x ∈ R gives values in (0, π).

Domain and Range of Inverse Trigonometric Functions Class-12th Board

Properties of Inverse Trigonometric Functions Class-12th Board

Understanding the properties of these functions is essential for solving equations and simplifying expressions. Here are some key properties:

Properties:

• sin^-1 (-x) = -sin^-1 (x)
• cos^-1 (-x) = π – cos^-1 (x)
• tan^-1 (-x) = -tan^-1 (-x)
• cosec⁻¹(−x) = − cosec⁻¹(x)
• sec⁻¹(−x) = π − sec⁻¹(x)
• cot⁻¹(−x) = π − cot⁻¹(x)

• sin⁻¹(x) = cosec⁻¹(1/x), x∈ [−1,1]−{0}
• cos⁻¹(x) = sec⁻¹(1/x), x ∈ [−1,1]−{0}
• tan⁻¹(x) = cot⁻¹(1/x), if x > 0  (or)  cot⁻¹(1/x) −π, if x < 0
• cot⁻¹(x) = tan⁻¹(1/x), if x > 0 (or) tan⁻¹(1/x) + π, if x < 0

• sin⁻¹(1/x) = cosec⁻¹x, x≥1 or x≤−1
• cos⁻¹(1/x) = sec⁻¹x, x≥1 or x≤−1
• tan⁻¹(1/x) = −π + cot⁻¹(x)

• Sin⁻¹(cos θ) = π/2 − θ, if θ∈[0,π]
• Cos⁻¹(sin θ) = π/2 − θ, if θ∈[−π/2, π/2]
• Tan⁻¹(cot θ) = π/2 − θ, θ∈[0,π]
• Cot⁻¹(tan θ) = π/2 − θ, θ∈[−π/2, π/2]
• Sec⁻¹(cosec θ) = π/2 − θ, θ∈[−π/2, 0]∪[0, π/2]
• Cosec⁻¹(sec θ) = π/2 − θ, θ∈[0,π]−{π/2}
• Sin⁻¹(x) = cos⁻¹[√(1−x²)], 0≤x≤1
• sin^-1 (x) = -cos⁻¹[√(1−x²)], −1≤x<0

• sin⁻¹(x) + sin⁻¹(y) = sin⁻¹[x√(1−y²)+ y√(1−x²)]
• cos⁻¹x + cos⁻¹y = cos⁻¹[xy−√(1−x²)√(1−y²)]

Addition and Subtraction Formula:

These formulas help in solving more complex problems:

sin^-1 x + cos^-1 x = π/2

tan^-1 x + cot^-1 x = π/2

Sec⁻¹x + Cosec⁻¹x = π/2

Other Important Identites:

• sin(sin^-1 x) = x, for x ∈ [-1, 1]
• cos(cos^-1 x) = x, for x ∈ [-1, 1]
• tan(tan^-1 x) = x, for x ∈ R
• cot(cot^-1 x) = x, for x ∈ R
• sec(sec^-1 x) = x, for x ∈ R – {0}
• cosec(cosec^-1 x) = x for x ∈ R – {0}

Also, the following formulas are defined for inverse trigonometric functionns:

• sin^-1 (siny) = y, for -π/2 ≤ y ≤ π/2
• cos⁻¹(cos y) =y, for 0 ≤ y ≤ π
• tan⁻¹(tan y) = y, for -π/2 <y< π/2
• cot⁻¹(cot y) = y for 0<y< π
• sec⁻¹(sec y) = y, for 0 ≤ y ≤ π, y ≠ π/2
• cosec⁻¹(cosec y) = y, for -π/2 ≤ y ≤ π/2, y ≠ 0
  

Graphs of Inverse Trigonometric Functions Class-12th Board

The graphs of inverse trigonometric functions help visualize the behavior of these functions.

y = sin^-1 x: This graph is confined between [-1, 1] on the x-axis and [-π/2, π/2] on the y-axis. It is an increasing function.

y = cos^-1 x: This graph is confined between [-1, 1] on the x-axis and [0, π] on the y-axis. It is an decreasing function.

y = tan^-1 x: This graph is spans the entire x-axis but is bounded between (- [-π/2, π/2] on the y-axis.

Solving Equations Inverse Trigonometric Functions Class-12th Board

To solve equations, students must apply the properties and identities of inverse trigonometric functions. For example:

sin^-1 x + cos^-1 x = π/2

tan^-1 [(a+b)/(1-ab)] = tan^-1 a + tan^-1 b for ab < 0.

Applications of Inverse Trigonometric Functions Class-12th Board

Inverse trigonometric functions have a wide range of applications, such as:

Solving integrals and differential equations in calculus.
Finding the angles in geometrical problems.
Modeling real-world phenomena in physics and engineering.

Summary

Inverse trigonometric functions are a critical concept in mathematics, with significant applications in both academic problems and practical scenarios. Mastery of their properties, domains, ranges, and applications will provide a solid foundation for further studies in calculus and related fields.

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