Inverse Trigonometry Revision

Important Points in 10 Minutes

Key Definitions

Trigonometric Functions

• Input: Angles (which we observe)
• Output: Numerical Value
• Domain: Values of input angles (x)
• Range: Output numerical values (y)

Inverse Trigonometric Functions

• Input: Numerical Value
• Output: Angle
• Example:
  • Sine Inverse
  • Cosine Inverse
  • Tangent Inverse

Key Conditions

• The domain's one value of x cannot have two values in the range
• One numerical value can't correspond to two angles
• The range of sine inverse has to be restricted

Important Ranges and Domains

• Range of Sine Inverse:
  • • π/2 to π/2 (Principal Value Branch)
• Principal Range of Cosine Inverse, Secant Inverse, Cosecant Inverse:
  • 0 to π
• Principal Range of Sine Inverse, Tangent Inverse, Cotangent Inverse:
  • -π/2 to π/2

Principal Value Branch:

• The smallest range of angles where no conflict occurs.
• Example: The principal value of sine inverse (0.5) will be π/6.

Important Formulas and Families

1. • Tangent Inverse (x/y) =
     • Tangent Inverse x + Tangent Inverse y if x * y < 1
2. • Sine Inverse (1/x) = Cosine Inverse (sqrt(x^2 - 1))
     • Condition: |x| > 1*

Question Solving Strategies

• Remember important conditions
• Use the correct principal value and principal branch
• Utilize correct formulae and families

Important Transformations

• For Cosine Inverse, Secant Inverse, and Cosecant Inverse, the minus sign cannot directly come out

Additional Important Formulas

1. • Sine Inverse x + Cosine Inverse x = π/2
2. • Tangent Inverse x + Cotangent Inverse x (and for other such combined formulas):
     • The sine (inside value should equal the final value)

Conclusion

• Remember these important points
• Be able to solve all difficult questions
• Happy Learning!