Inverse Trigonometric Functions
Inverse Trigonometric Functions
This domain checks principal values, identities, and the ability to undo a trigonometric function only inside its chosen branch. UPMSP questions often combine a value with a domain restriction or ask for a simplified expression.
Core rules
- sin^{-1}x: domain [-1,1], range [-π/2,π/2].
- cos^{-1}x: domain [-1,1], range [0,π].
- tan^{-1}x: domain R, range (-π/2,π/2).
- The principal value is the unique value in the stated range, not every angle with that sine or cosine.
- sin^{-1}x+cos^{-1}x=π/2 for x∈[-1,1].
- tan^{-1}x+tan^{-1}y=tan^{-1}((x+y)/(1-xy)) when xy<1; if xy>1, add or subtract π according to signs.
- 2tan^{-1}x=tan^{-1}(2x/(1-x^2)) for |x|<1.
Worked example
Evaluate sin^{-1}(sin(5π/6)). The angle 5π/6 is outside [-π/2,π/2]. Simplify first: sin(5π/6)=1/2. Then sin^{-1}(1/2)=π/6, because π/6 lies in the principal range. Writing 5π/6 directly is the common error.
Traps
Ignoring branch ranges; cancelling sin^{-1}(sin x)=x outside the range; using the tan addition formula when xy>1 without the π adjustment; mixing degrees and radians; forgetting that cos^{-1} has range [0,π], not [-π/2,π/2].