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Inverse Trigonometric Functions

Inverse trigonometric functions are used to find angles when a trigonometric ratio is known. Since ordinary trigonometric functions are many-one on their natural domains, each inverse function is defined only after choosing a principal value branch. The chapter begins with domains and ranges of sin⁻¹x, cos⁻¹x, tan⁻¹x, cot⁻¹x, sec⁻¹x and cosec⁻¹x. These intervals are not optional details; they decide whether an answer such as π/6, 5π/6, −π/6 or 7π/6 is acceptable. A major exam focus is simplification using identities such as sin⁻¹x + cos⁻¹x = π/2 and tan⁻¹x + cot⁻¹x = π/2. Students must check the domain of the variable and the range of the final inverse function before applying a formula. Sum and difference formulae, especially for tan⁻¹x, are useful in reducing expressions and solving equations. The condition xy < 1 or xy > 1 affects whether an additional π adjustment is needed, so branch checking is essential. Graph-based understanding helps students remember why branches are restricted. The graph of an inverse function is obtained by reflecting the restricted original function in the line y = x, with domain and range interchanged.