Bihar Board - Class 12 Math - Chapter 2: Inverse Trigonometric Function Handwritten Notes

Inverse trigonometric functions are the reverse of the basic trigonometric functions. While trigonometric functions such as sine, cosine, tangent, etc., take an angle as input and give a ratio as output, inverse trigonometric functions do the opposite. They take a ratio as input and give an angle as output. This chapter focuses on the definitions, properties, and applications of inverse trigonometric functions, which are essential in solving trigonometric equations and understanding relationships in geometry.

Key Points:

1. Definition of Inverse Trigonometric Functions:

   • The inverse of sine is arcsin or sin⁻¹.
   • The inverse of cosine is arccos or cos⁻¹.
   • The inverse of tangent is arctan or tan⁻¹.
   • The inverse of cotangent is arccot or cot⁻¹.
   • The inverse of secant is arcsec or sec⁻¹.
   • The inverse of cosecant is arccsc or csc⁻¹.

2. Principal Value Branch:

   • The principal value of an inverse trigonometric function is the unique value of the angle that lies within a specific range:
     • arcsin(x): Range $$[−π/2,π/2][-π/2, π/2]
     • arccos(x): Range $$[0,π][0, π]
     • arctan(x): Range $$[−π/2,π/2][-π/2, π/2]
     • Other inverse functions have their own ranges.

3. Properties of Inverse Trigonometric Functions:

   • If $$y=sin⁡−1(x)y = \sin^{-1}(x), then $$sin⁡(y)=x\sin(y) = x and $$yy lies within $$[−π/2,π/2][-π/2, π/2].
   • The inverse trigonometric functions are continuous within their domains and have specific ranges to ensure that they are functions (i.e., each input corresponds to exactly one output).
   • Domain:
     • arcsin(x) and arccos(x) have a domain of $$[−1,1][-1, 1].
     • arctan(x), arccot(x), arcsec(x), and arccsc(x) have domains of all real numbers except points that make the denominator zero.

4. Graphs of Inverse Trigonometric Functions:

   • The graphs of inverse trigonometric functions are reflections of the corresponding trigonometric functions over the line $$y=xy = x.
   • These graphs help to visualize the principal values and the domains/ranges of inverse functions.

5. Use of Inverse Trigonometric Functions:

   • Inverse trigonometric functions are widely used to find angles when the values of trigonometric ratios are known.
   • They are important for solving equations involving trigonometric ratios and are often used in integration, geometry, and calculus problems.

6. Composite Functions Involving Inverse Trigonometric Functions:

   • When inverse trigonometric functions are used in compositions with other functions, special attention must be paid to the domains and ranges to ensure correct results.
   • Example: $$sin⁡(cos⁡−1(x))\sin(\cos^{-1}(x)) or $$tan⁡(sin⁡−1(x))\tan(\sin^{-1}(x)).

Conclusion:

Inverse trigonometric functions are vital in mathematics, especially in solving trigonometric equations and understanding geometric relationships. Their proper use helps find angles when trigonometric ratios are known, and they are fundamental in calculus and physics. The key to mastering inverse trigonometric functions is understanding their definitions, properties, and the correct domain and range of each function. These functions simplify complex problems, making them essential tools in both theoretical and applied mathematics.