Inverse Trigonometric Functions - Detailed Notes

1. Principal Value and General Solution:

• Principal Value:
• The principal value of an inverse trigonometric function is the unique value within its principal range.
• For inverse sine (sin^-1 x), principal range = [-π/2, π/2].
• For inverse cosine (cos^-1 x), principal range = [0, π].
• For inverse tangent (tan^-1 x), principal range = (-π/2, π/2).
• General Solution:
• The general solution of an inverse trigonometric function includes all possible solutions obtained by adding or subtracting integer multiples of 2π (for sine and cosine) or π (for tangent) to the principal value.

2. Properties and Identities:

• Domain and Range:

• Domain: For sin^-1 x, [-1, 1]; cos^-1 x, [-1, 1]; tan^-1 x, R (all real numbers).

• Range: For sin^-1 x, [-π/2, π/2]; cos^-1 x, [0, π]; tan^-1 x, (-π/2, π/2).

• Even/Odd Functions:

• sin^-1 x and tan^-1 x are odd functions.

• cos^-1 x is an even function.

• Periodicity:

• sin^-1 x and tan^-1 x have a period of 2π.

• cos^-1 x has a period of 2π.

3. Graphs:

• Graphs of sin^-1 x and tan^-1 x:

• Both graphs pass through the origin.

• sin^-1 x is an increasing function and goes from -π/2 to π/2 as x increases from -1 to 1.

• tan^-1 x is also an increasing function and goes from -π/2 to π/2 as x increases from -∞ to ∞.

• Graph of cos^-1 x:

• Graph of cos^-1 x lies in the interval [0, π].

• As x increases from - 1 to 1, the graph starts at π and decreases to 0.

4. Composition of Functions:

• Identities:

• sin(cos^-1 x) = sqrt(1 - x^2) for -1 ≤ x ≤ 1

• cos(sin^-1 x) = sqrt(1 - x^2) for -1 ≤ x ≤ 1

• tan(cos^-1 x) = sqrt(1 - x^2) / x for 0 ≤ x ≤ 1, x ≠ 0

• Composition of Functions:

• Compositions with trigonometric functions can yield simplified expressions or identities.

5. Solving Equations:

• Solving Trigonometric Equations:
• Inverse trigonometric functions are used to solve trigonometric equations.
• For example, to solve the equation sin x = 0.5, we can find the value of x using the inverse sine function: x = sin^-1 (0.5) = π/6.

6. Applications:

• Finding Angles:
• Inverse trigonometric functions are used in finding angles of elevation or depression in real-life measurements and surveying.
• Oblique Triangles:
• Inverse trigonometric functions are employed in solving problems related to oblique triangles, where not all angles are right angles.
• Modeling Periodic Phenomena:
• Used in modeling periodic functions like sinusoidal motion and oscillatory behavior.

7. Integration and Differentiation:

• Integration:

• ∫ sin^-1 x dx = x sin^-1 x + sqrt(1 - x^2) + C

• ∫ cos^-1 x dx = x cos^-1 x - sqrt(1 - x^2) + C

• ∫ tan^-1 x dx = x tan^-1 x - ½ ln | sec x + tan x | + C

• Differentiation:

• d/dx sin^-1 x = 1/sqrt(1 - x^2)

• d/dx cos^-1 x = -1/sqrt(1 - x^2)

• d/dx tan^-1 x = 1/(1 + x^2)

8. Parametric Equations and Curves:

• Parametric Equations:
• Inverse trigonometric functions can be used to define parametric equations of curves, representing their motion or trajectory.

9. Inverse Hyperbolic Functions:

• Definition and Properties:
• Analogous to inverse trigonometric functions but involve hyperbolic functions.
• Properties similar to inverse trigonometric functions.

10. Trigonometric Equations and Inequalities:

• Solving Advanced Equations and Inequalities:
• Use inverse trigonometric functions to solve complex trigonometric equations and inequalities, such as equations with multiple angles or solutions.

References:

• NCERT Mathematics Textbook Class 11, Chapter 3: Trigonometric Functions
• NCERT Mathematics Textbook Class 12, Chapter 2: Inverse Trigonometric Functions