Recap of Inverse Trigonometric Functions

• Definition: Inverse function of a trigonometric function
• Range of inverse trig functions

Domain and Range of Inverse Trigonometric Functions

• Domain: Range of corresponding trig functions
• Range: Restricted domains of inverse trig functions

Basic Properties of Inverse Trig Functions

• Identity Property: sin^(-1)(sin(x)) = x
• Identity Property: cos^(-1)(cos(x)) = x
• Identity Property: tan^(-1)(tan(x)) = x

Odd and Even Properties of Inverse Trig Functions

• Odd Property: sin^(-1)(-x) = -sin^(-1)(x)
• Even Property: cos^(-1)(-x) = cos^(-1)(x)
• Odd Property: tan^(-1)(-x) = -tan^(-1)(x)

Pythagorean Identities of Inverse Trig Functions

• Pythagorean Identity: sin^(-1)(x) = cos^(-1)(√(1 - x^2))
• Pythagorean Identity: cos^(-1)(x) = sin^(-1)(√(1 - x^2))

Examples: Identifying Domain and Range

• Example 1: Find the domain and range of sin^(-1)(x)
• Example 2: Find the domain and range of cos^(-1)(x)

Complementary Angles Property

• Complementary Angles Property: sin^(-1)(x) + cos^(-1)(x) = π/2
• Complementary Angles Property: tan^(-1)(x) + cot^(-1)(x) = π/2

Examples: Complementary Angles Property

• Example 1: Find the value of sin^(-1)(3/5) + cos^(-1)(4/5)
• Example 2: Find the value of tan^(-1)(2) + cot^(-1)(3)

Composition of Inverse Trig Functions

• Composition Property: sin(sin^(-1)(x)) = x
• Composition Property: cos(cos^(-1)(x)) = x
• Composition Property: tan(tan^(-1)(x)) = x

Examples: Composition Property

• Example 1: Prove that sin(sin^(-1)(x)) = x
• Example 2: Prove that tan(tan^(-1)(x)) = x

• Example 1: Find the domain and range of sin^(-1)(x)
  • Domain: -1 ≤ x ≤ 1 (range of sin function)
  • Range: -π/2 ≤ sin^(-1)(x) ≤ π/2 (restricted range)
• Example 2: Find the domain and range of cos^(-1)(x)
  • Domain: -1 ≤ x ≤ 1 (range of cos function)
  • Range: 0 ≤ cos^(-1)(x) ≤ π (restricted range)

• Complementary Angles Property: sin^(-1)(x) + cos^(-1)(x) = π/2
  • The sine and cosine of complementary angles always add up to π/2
  • Useful in finding missing angles in right-angled triangles
• Complementary Angles Property: tan^(-1)(x) + cot^(-1)(x) = π/2
  • The tangent and cotangent of complementary angles always add up to π/2
  • Used to find missing angles in right-angled triangles

• Example 1: Find the value of sin^(-1)(3/5) + cos^(-1)(4/5)
  • Using the Complementary Angles Property: sin^(-1)(3/5) + cos^(-1)(4/5) = π/2
• Example 2: Find the value of tan^(-1)(2) + cot^(-1)(3)
  • Using the Complementary Angles Property: tan^(-1)(2) + cot^(-1)(3) = π/2

• Composition Property: sin(sin^(-1)(x)) = x
  • The sine of the arcsine of x is x itself
• Composition Property: cos(cos^(-1)(x)) = x
  • The cosine of the arccosine of x is x itself
• Composition Property: tan(tan^(-1)(x)) = x
  • The tangent of the arctangent of x is x itself

• Example 1: Prove that sin(sin^(-1)(x)) = x
  • Let y = sin^(-1)(x)
  • Then, sin(y) = x (definition of arcsine)
  • Taking the sine of both sides, sin(sin^(-1)(x)) = x
• Example 2: Prove that tan(tan^(-1)(x)) = x
  • Let y = tan^(-1)(x)
  • Then, tan(y) = x (definition of arctangent)
  • Taking the tangent of both sides, tan(tan^(-1)(x)) = x

Derivatives of Inverse Trig Functions

• Derivative of arcsin(x) = 1/√(1 - x^2)
• Derivative of arccos(x) = -1/√(1 - x^2)
• Derivative of arctan(x) = 1/(1 + x^2)

Integrals of Inverse Trig Functions

• Integral of 1/√(1 - x^2) = arcsin(x) + C
• Integral of -1/√(1 - x^2) = arccos(x) + C
• Integral of 1/(1 + x^2) = arctan(x) + C

Logarithmic Properties of Inverse Trig Functions

• Logarithmic Property: ln(1 + x) = arctan(x) + C
• Logarithmic Property: ln(1 - x) = -arctan(x) + C

Properties of Inverse Hyperbolic Trig Functions

• Inverse Hyperbolic Functions: arcsinh(x), arccosh(x), arctanh(x)
• Similar properties as inverse trig functions

• Derivative of arcsin(x) = 1/√(1 - x^2)
  • Example: Find the derivative of y = arcsin(2x)
• Derivative of arccos(x) = -1/√(1 - x^2)
  • Example: Find the derivative of y = arccos(3x)
• Derivative of arctan(x) = 1/(1 + x^2)
  • Example: Find the derivative of y = arctan(4x)

• Integral of 1/√(1 - x^2) = arcsin(x) + C
  • Example: Evaluate the integral ∫(1/√(1 - x^2)) dx
• Integral of -1/√(1 - x^2) = arccos(x) + C
  • Example: Evaluate the integral ∫(-1/√(1 - x^2)) dx
• Integral of 1/(1 + x^2) = arctan(x) + C
  • Example: Evaluate the integral ∫(1/(1 + x^2)) dx

• Logarithmic Property: ln(1 + x) = arctan(x) + C
  • Example: Prove that ln(1 + x) = arctan(x) + C
• Logarithmic Property: ln(1 - x) = -arctan(x) + C
  • Example: Prove that ln(1 - x) = -arctan(x) + C

• Inverse Hyperbolic Functions: arcsinh(x), arccosh(x), arctanh(x)
  • Similar to inverse trig functions, but for hyperbolic functions
  • Inverse Hyperbolic Identity: sinh(arcsinh(x)) = x
  • Inverse Hyperbolic Identity: cosh(arccosh(x)) = x
  • Inverse Hyperbolic Identity: tanh(arctanh(x)) = x

• Inverse Hyperbolic Identity: arcsinh(sinh(x)) = x
• Inverse Hyperbolic Identity: arccosh(cosh(x)) = x, x ≥ 1
• Inverse Hyperbolic Identity: arccosh(cosh(x)) = -x, x ≤ -1
• Inverse Hyperbolic Identity: arctanh(tanh(x)) = x

• Graph of y = sin^(-1)(x)
• Graph of y = cos^(-1)(x)
• Graph of y = tan^(-1)(x)

• Graph of y = sinh^(-1)(x)
• Graph of y = cosh^(-1)(x)
• Graph of y = tanh^(-1)(x)

• Method: Convert inverse trig equation to a regular trig equation
• Example: Solve the equation sin^(-1)(x) + cos^(-1)(x) = π/2
• Example: Solve the equation tan^(-1)(x) = π/4

• Inverse Trig Functions: arcsin(x), arccos(x), arctan(x)
• Properties of Inverse Trig Functions
• Derivatives and Integrals of Inverse Trig Functions
• Logarithmic Properties of Inverse Trig Functions
• Properties of Inverse Hyperbolic Trig Functions
• Solving Equations Involving Inverse Trig Functions
• Graphs of Inverse Trig Functions and Inverse Hyperbolic Trig Functions