Inverse Trigonometric Functions

 Inverse Trigonometric Functions

The Inverse trigonometric functions are the inverse functions of the trigonometric functions (with well-defined domains). Specifically, they are inverses of sine, cosine, tangent, cotangent, secant, and cosecant functions. They are used to find an angle in any trigonometric measurement of an angle.

1. Introduction:

   • Trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent
   • Domain, range, and periodicity of trigonometric functions
   • Need for inverse trigonometric functions

2. Definition of Inverse Trigonometric Functions:

   • Inverse of a function and its notation
   • Domain and range of inverse trigonometric functions

3. Inverse Trigonometric Functions:

   • Inverse sine function (sin^(-1)(x))
   • Inverse cosine function (cos^(-1)(x))
   • Inverse tangent function (tan^(-1)(x))
   • Inverse cosecant function (cosec^(-1)(x))
   • Inverse secant function (sec^(-1)(x))
   • Inverse cotangent function (cot^(-1)(x))

4. Principal Values and Ranges:

   • Principal values of inverse trigonometric functions
   • Ranges of inverse trigonometric functions

5. Properties and Identities:

   • Relationships between trigonometric functions and their inverses
   • Properties and identities involving inverse trigonometric functions

6. Graphs of Inverse Trigonometric Functions:

   • Graphs of inverse sine, inverse cosine, and inverse tangent functions
   • Domain, range, and behavior of graphs

7. Evaluation of Inverse Trigonometric Functions:

   • Evaluating inverse trigonometric functions using special triangles and unit circle
   • Evaluating inverse trigonometric functions using identities and properties

8. Solving Equations and Problems:

   • Solving trigonometric equations using inverse trigonometric functions
   • Applications of inverse trigonometric functions in real-world problems

1. Property of Inverse Sine Function (sin^(-1)(x)):

   • Domain: [-1, 1]
   • Range: [-π/2, π/2]
   • sin(sin^(-1)(x)) = x for all x in the domain
   • sin^(-1)(sin(x)) = x for all x in the range

2. Property of Inverse Cosine Function (cos^(-1)(x)):

   • Domain: [-1, 1]
   • Range: [0, π]
   • cos(cos^(-1)(x)) = x for all x in the domain
   • cos^(-1)(cos(x)) = x for all x in the range

3. Property of Inverse Tangent Function (tan^(-1)(x)):

   • Domain: (-∞, +∞)
   • Range: (-π/2, π/2)
   • tan(tan^(-1)(x)) = x for all x in the domain
   • tan^(-1)(tan(x)) = x for all x in the range

4. Property of Inverse Cosecant Function (cosec^(-1)(x)):

   • Domain: [-∞, -1] ∪ [1, +∞]
   • Range: [-π/2, 0] ∪ [0, π/2]
   • cosec(cosec^(-1)(x)) = x for all x in the domain
   • cosec^(-1)(cosec(x)) = x for all x in the range

5. Property of Inverse Secant Function (sec^(-1)(x)):

   • Domain: [-∞, -1] ∪ [1, +∞]
   • Range: [0, π/2] ∪ [π/2, π]
   • sec(sec^(-1)(x)) = x for all x in the domain
   • sec^(-1)(sec(x)) = x for all x in the range

6. Property of Inverse Cotangent Function (cot^(-1)(x)):

   • Domain: (-∞, +∞)
   • Range: (0, π)
   • cot(cot^(-1)(x)) = x for all x in the domain
   • cot^(-1)(cot(x)) = x for all x in the range

1. Theorem: sin^(-1)(x) + cos^(-1)(x) = π/2

   • This theorem states that the sum of the inverse sine of an angle and the inverse cosine of the same angle is equal to π/2.

2. Theorem: tan^(-1)(x) + cot^(-1)(x) = π/2

   • This theorem states that the sum of the inverse tangent of an angle and the inverse cotangent of the same angle is equal to π/2.

3. Theorem: sin^(-1)(x) = cos^(-1)(√(1-x^2))

   • This theorem relates the inverse sine of an angle to the inverse cosine of the complementary angle. It states that the inverse sine of x is equal to the inverse cosine of the square root of (1 - x^2).

4. Theorem: cos^(-1)(x) = sin^(-1)(√(1-x^2))

   • This theorem is similar to the previous one but relates the inverse cosine of an angle to the inverse sine of the complementary angle. It states that the inverse cosine of x is equal to the inverse sine of the square root of (1 - x^2).

5. Theorem: tan^(-1)(x) = cot^(-1)(1/x)

   • This theorem relates the inverse tangent of an angle to the inverse cotangent of the reciprocal of the angle. It states that the inverse tangent of x is equal to the inverse cotangent of 1/x.

6. Theorem: sin^(-1)(x) + sin^(-1)(y) = sin^(-1)(x√(1-y^2) + y√(1-x^2))

   • This theorem provides a formula for the sum of two inverse sine values. It states that the sum of the inverse sine of x and the inverse sine of y is equal to the inverse sine of the sum of (x√(1-y^2)) and (y√(1-x^2)).

• WORK SHEET 1
• WORK SHEET 2
• WORK SHEET 3
• MCQ WORK SHEET 1
• MCQ WORK SHEET 2
• MCQ WORK SHEET 3
• CHAPTER TEST 1
• CHAPTER TEST 2
• CHAPTER TEST 3