Inverse Trigonometric Functions
Inverse Trigonometric Functions tests principal-value branches, domains, identities, and solution of equations. The key is that each inverse function returns exactly one angle in a prescribed interval, not every angle with the same sine or tangent.
Core rules
sin⁻¹x has domain [−1,1] and range [−π/2,π/2]; cos⁻¹x has range [0,π]; tan⁻¹x has range (−π/2,π/2). Also cot⁻¹x has range (0,π), sec⁻¹x has range [0,π] except π/2, and csc⁻¹x has range [−π/2,π/2] except 0. Standard identities include sin⁻¹x+cos⁻¹x=π/2 and tan⁻¹x+cot⁻¹x=π/2. The addition formula tan⁻¹x+tan⁻¹y=tan⁻¹((x+y)/(1−xy)) needs a correction of +π or −π when xy>1, according to the quadrant.
Worked example
Evaluate tan⁻¹1+tan⁻¹2+tan⁻¹3. First tan⁻¹1=π/4. For tan⁻¹1+tan⁻¹2, xy=2>1, so the sum lies in (π/2,π) and equals π+tan⁻¹((1+2)/(1−2))=π−tan⁻¹3. Adding tan⁻¹3 gives π.
Traps
Do not cancel sin⁻¹(sin x) to x unless x lies in the principal range. Check domains before applying identities; for example tan⁻¹x+tan⁻¹(1/x) is not always π/2. In equations, reject extraneous angles outside the principal branch. Remember that the range of sec⁻¹ and csc⁻¹ excludes values where the reciprocal is undefined.