Inverse Trigonometric Functions - Domain, Range, and Graph of tan inverse

Overview

  • Inverse Trigonometric Functions
  • Definition and Notation
  • Concept of Domain and Range
  • Graph of tan inverse
  • Application in problem-solving

Inverse Trigonometric Functions

  • Inverse of a trigonometric function
  • Denoted as f-1(x)
  • Returns the original angle when given the ratio or value

Definition and Notation

  • tan-1(x) is the inverse of the tangent function
  • Can also be written as atan(x) or arctan(x)

Domain and Range of tan inverse

  • Domain: (-∞, ∞)
  • Range: (-π/2, π/2)

Graph of tan inverse

  • X-axis represents input values
  • Y-axis represents output values
  • Asymptote at x = -∞ and x = ∞
  • Function is not defined at x = ±π/2

Graph of tan inverse (cont.)

  • Graph is symmetric with respect to the origin
  • Increasing and decreasing portions
  • Infinite number of turning points

Graph of tan inverse (cont.)

Graph of tan inverse

Application in problem-solving

  • Finding unknown angles in triangles
  • Evaluating trigonometric ratios for specific values
  • Solving equations involving trigonometric functions

Example 1

Find the value of tan inverse(1).

  • Solution:
    • tan inverse(1) = π/4

Example 2

Find the value of tan inverse(0).

  • Solution:
    • tan inverse(0) = 0

Example 3

Find the value of tan inverse(-1).

  • Solution:
    • tan inverse(-1) = -π/4

Example 4

Evaluate the expression:

  • tan inverse(√3).
  • Solution:
    • tan inverse(√3) = π/3

Example 5

Evaluate the expression:

  • tan inverse(-√3).
  • Solution:
    • tan inverse(-√3) = -π/3

Important properties of tan inverse

  1. tan inverse(x + y) = tan inverse(x) + tan inverse(y)
  1. tan inverse(x - y) = tan inverse(x) - tan inverse(y)
  1. tan inverse(1/x) = π/2 - tan inverse(x)
  1. tan inverse(x) + tan inverse(1/x) = π/2

Important properties of tan inverse (cont.)

  1. tan inverse(x) + tan inverse(-x) = 0
  1. tan inverse(x) - tan inverse(-x) = π/2
  1. tan inverse(x) + tan inverse(1/x) = π/2
  1. tan inverse(i) = i tan inverse(1)
    • where i is the imaginary unit and tan inverse(1) = π/4

Derivatives of tan inverse

  • d/dx(tan inverse(x)) = 1/(1 + x^2)
  • d/dx(tan inverse(1/x)) = -1/(x^2 + 1)

Integration of tan inverse

  • ∫(1/(x^2 + a^2)) dx = (1/a) tan inverse(x/a) + C

Summary

  • Inverse Trigonometric Functions return the original angle for a given ratio or value
  • The domain of tan inverse is (-∞, ∞) and the range is (-π/2, π/2)
  • The graph of tan inverse is symmetric with respect to the origin, with infinite turning points
  • tan inverse has several important properties regarding addition, subtraction, and reciprocals
  • The derivative of tan inverse is 1/(1 + x^2), and the integral of 1/(x^2 + a^2) is (1/a) tan inverse(x/a) + C

Practice Problems

  1. Find the value of tan inverse(√2).
  1. Evaluate the expression: tan inverse(-1/√3).
  1. Differentiate the function f(x) = tan inverse(x).
  1. Integrate the function g(x) = 1/(5x^2 + 3).
  1. Solve the equation: tan inverse(x) = -π/6.

Solving Problems Involving Inverse Trigonometric Functions

  • Substitution: Replace inverse trigonometric functions with appropriate variables
  • Simplify: Use trigonometric identities to simplify the equation
  • Solve: Solve the resulting equation for the variable
  • Substitute: Substitute the values back to find the original angle

Example 1

Solve the equation: sin^(-1)(2x + 1) = π/6.

  • Solution:
    • Let 2x + 1 = sin(π/6)
    • Simplify: 2x + 1 = 1/2
    • Solve: 2x = 1/2 - 1 = -1/2
    • x = -1/4
    • Substitute back: sin^(-1)(2(-1/4) + 1) = π/6, which is true.

Example 2

Solve the equation: cos^(-1)(1 - 2x) = 3π/4.

  • Solution:
    • Let 1 - 2x = cos(3π/4)
    • Simplify: 1 - 2x = -1/√2
    • Solve: 2x = 1 + 1/√2 = (√2 + 1)/√2
    • x = (√2 + 1)/(2√2)
    • Substitute back: cos^(-1)(1 - 2((√2 + 1)/(2√2))) = 3π/4, which is true.

Example 3

Solve the equation: tan^(-1)(2 - x) + tan^(-1)(3x) = π/2.

  • Solution:
    • Let 2 - x = tan(π/4)
    • Simplify: 2 - x = 1
    • Solve: x = 2 - 1 = 1
    • Substitute back: tan^(-1)(2 - 1) + tan^(-1)(3(1)) = π/2, which is true.

Example 4

Solve the equation: sin^(-1)(x) + cos^(-1)(x) = π/4.

  • Solution:
    • Let x = sin(π/6) = cos(π/3) = 1/2
    • Substitute back: sin^(-1)(1/2) + cos^(-1)(1/2) = π/4, which is true.

Example 5

Solve the equation: tan^(-1)(3x) = cos^(-1)(5x).

  • Solution:
    • Let 3x = tan(π/3)
    • Simplify: 3x = √3
    • Solve: x = (√3)/3
    • Substitute back: tan^(-1)(3((√3)/3)) = cos^(-1)(5((√3)/3)), which is true.

Integration of Inverse Trigonometric Functions

  • Common integrals:
    • ∫(1/(a^2 + x^2)) dx = (1/a) tan^(-1)(x/a) + C
    • ∫(1/(a^2 - x^2)) dx = (1/(2a)) ln|((a+x)/(a-x))| + C

Example 1

Evaluate ∫(1/(4 + x^2)) dx.

  • Solution:
    • Using the formula, the integral is (1/4) tan^(-1)(x/2) + C

Example 2

Evaluate ∫(1/(3 - x^2)) dx.

  • Solution:
    • Using the formula, the integral is (1/6) ln|((3+x)/(3-x))| + C

Summary

  • Solving problems involving inverse trigonometric functions requires substitution and simplification
  • Integration of inverse trigonometric functions can be done using specific formulas
  • Practice various examples to strengthen problem-solving skills