• sin^-1(x) as an angle whose sine is x
• cos^-1(x) as an angle whose cosine is x
• tan^-1(x) as an angle whose tangent is x

Integrating inverse trigonometric functions

Integrals of sin^-1(x), cos^-1(x), and tan^-1(x)

Approaches to integrating inverse trigonometric functions

Evaluating indefinite integrals and solving definite integrals Inverse Trigonometric Functions - Difference of two inverses of tangent

Definition of inverse tangent function: tan^-1(x) as an angle whose tangent is x

Let’s consider two angles, A and B, such that tan(A) = p and tan(B) = q

Difference between the two angles can be expressed as: A - B = tan^-1(p) - tan^-1(q)

Using the trigonometric identity tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A) tan(B))

Substituting the values, we get: A - B = tan^-1((p - q) / (1 + pq))

This formula gives us the difference between two angles whose tangents are given

Similar to the difference formula, we can also derive the formula for addition of two inverses of tangent

Let A and B be two angles such that tan(A) = p and tan(B) = q

The sum of the two angles can be expressed as: A + B = tan^-1(p) + tan^-1(q)

Using the trigonometric identity tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A) tan(B))

Substituting the values, we get: A + B = tan^-1((p + q) / (1 - pq))

This formula gives us the sum of two angles whose tangents are given

Cotangent is the reciprocal of tangent, i.e., cot(x) = 1 / tan(x)

We can express the inverse trigonometric functions of tangent in terms of cotangent

The expressions are as follows:

• cot^-1(x) = tan^-1(1 / x)
• tan^-1(x) = cot^-1(1 / x)
• cot^-1(x) = π/2 - tan^-1(x)
• tan^-1(x) = π/2 - cot^-1(x)

  Example:
  Simplify the expression cot-1(√3)
  Solution:
  Using the formula, we have: cot-1(√3) = tan-1(1 / √3)
  Simplifying further, we get: cot-1(√3) = tan-1(√3/3)
  Inverse Trigonometric Functions - Expressions involving multiple angles

Inverse trigonometric functions can have expressions involving multiple angles

Let’s consider the formula for arcsin:

• sin(A + B) = sin(A) cos(B) + cos(A) sin(B)
• Replacing A and B with inverse sine functions, we get:
  • sin(sin^-1(x) + sin^-1(y)) = xy + √(1 - x^2) √(1 - y^2)

Similar formulas can be derived for other inverse trigonometric functions as well

The domain of inverse trigonometric functions is restricted to ensure one-to-oneness

As a result, the range of inverse trigonometric functions is limited

The range of inverse sine function (sin^-1(x)) is [-π/2, π/2]

The range of inverse cosine function (cos^-1(x)) is [0, π]

The range of inverse tangent function (tan^-1(x)) is [-π/2, π/2]

These ranges determine the maximum and minimum values that the inverse trigonometric functions can take

Graphs of inverse trigonometric functions help visualize their behavior

Graph of y = sin^-1(x):

• Domain: [-1, 1]
• Range: [-π/2, π/2]
• As x approaches -1 or 1, the corresponding y values approach -π/2 and π/2 respectively
• The graph is symmetric about the line y = x

Graphs of cos^-1(x) and tan^-1(x) have similar characteristics

Definition of inverse tangent function: tan^-1(x) as an angle whose tangent is x

Let’s consider two angles, A and B, such that tan(A) = p and tan(B) = q

Difference between the two angles can be expressed as: A - B = tan^-1(p) - tan^-1(q)

Using the trigonometric identity tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A) tan(B))

Substituting the values, we get: A - B = tan^-1((p - q) / (1 + pq))

This formula gives us the difference between two angles whose tangents are given

Similar to the difference formula, we can also derive the formula for addition of two inverses of tangent

Let A and B be two angles such that tan(A) = p and tan(B) = q

The sum of the two angles can be expressed as: A + B = tan^-1(p) + tan^-1(q)

Using the trigonometric identity tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A) tan(B))

Substituting the values, we get: A + B = tan^-1((p + q) / (1 - pq))

This formula gives us the sum of two angles whose tangents are given

Cotangent is the reciprocal of tangent, i.e., cot(x) = 1 / tan(x)

We can express the inverse trigonometric functions of tangent in terms of cotangent

The expressions are as follows:

• cot^-1(x) = tan^-1(1 / x)
• tan^-1(x) = cot^-1(1 / x)
• cot^-1(x) = π/2 - tan^-1(x)
• tan^-1(x) = π/2 - cot^-1(x)

  Example:
  Simplify the expression cot-1(√3)
  Solution:
  Using the formula, we have: cot-1(√3) = tan-1(1 / √3)
  Simplifying further, we get: cot-1(√3) = tan-1(√3/3)
  Inverse Trigonometric Functions - Expressions involving multiple angles

Inverse trigonometric functions can have expressions involving multiple angles

Let’s consider the formula for arcsin:

• sin(A + B) = sin(A) cos(B) + cos(A) sin(B)
• Replacing A and B with inverse sine functions, we get:
  • sin(sin^-1(x) + sin^-1(y)) = xy + √(1 - x^2) √(1 - y^2)

Similar formulas can be derived for other inverse trigonometric functions as well

The domain of inverse trigonometric functions is restricted to ensure one-to-oneness

As a result, the range of inverse trigonometric functions is limited

The range of inverse sine function (sin^-1(x)) is [-π/2, π/2]

The range of inverse cosine function (cos^-1(x)) is [0, π]

The range of inverse tangent function (tan^-1(x)) is [-π/2, π/2]

These ranges determine the maximum and minimum values that the inverse trigonometric functions can take

Graphs of inverse trigonometric functions help visualize their behavior

Graph of y = sin^-1(x):

• Domain: [-1, 1]
• Range: [-π/2, π/2]
• As x approaches -1 or 1, the corresponding y values approach -π/2 and π/2 respectively
• The graph is symmetric about the line y = x

Graphs of cos^-1(x) and tan^-1(x) have similar characteristics