Inverse Trigonometric Functions
Inverse Trigonometric Functions tests principal values, domains and ranges, identities, and the solution of equations involving inverse functions. The exam frequently uses the principal-range restrictions to decide between two possible angles.
Principal ranges
sin^{-1}x has range [−π/2,π/2]; cos^{-1}x has range [0,π]; tan^{-1}x has range (−π/2,π/2); cot^{-1}x has range (0,π); sec^{-1}x has range [0,π] excluding π/2; cosec^{-1}x has range [−π/2,π/2] excluding 0. Standard identities include sin^{-1}x+cos^{-1}x=π/2 and tan^{-1}x+cot^{-1}x=π/2. The sum formula tan^{-1}x+tan^{-1}y=tan^{-1}[(x+y)/(1−xy)] is valid only when xy<1; if xy>1, add or subtract π according to the sign of x and y. Similar restrictions apply to sin^{-1}x+sin^{-1}y.
Worked example
Find tan^{-1}1+tan^{-1}2+tan^{-1}3. For the first two, (1+2)/(1−1·2)=−3, so their sum has tangent −3 and lies in (π/2,π). Adding tan^{-1}3, the tangent is (−3+3)/(1−(−3)(3))=0. The sum is therefore 0 or π. Each angle is positive and the total exceeds π/2, so the value is π.
Traps
The formula giving tan=0 does not by itself decide between 0 and π; use the range and signs. Do not confuse sin^{-1}x with (sin x)^{-1}. sin^{-1}(sin x) equals x only when x lies in the principal range. When solving an equation, check every solution against the domains of the inverse functions.