Inverse Trigonometric Functions
Inverse Trigonometric Functions reverses sine, cosine and tangent only after restricting domains so that each inverse is a function. The exam tests principal values, domains and ranges, identities and simple equations.
Core rules
- sin inverse x has domain [-1,1] and range [-π/2,π/2]; cos inverse x has range [0,π]; tan inverse x has range (-π/2,π/2).
- sin inverse(sin x)=x only when x lies in the principal interval; otherwise reduce to that interval.
- sin inverse x + cos inverse x=π/2 for x in [-1,1].
- tan inverse x + tan inverse y=tan inverse((x+y)/(1-xy)) when xy<1; adjust by π when xy>1 and x,y positive.
Worked example
Evaluate sin inverse(sin 5π/6). The angle 5π/6 is not in [-π/2,π/2]. Since sin(5π/6)=sin(π/6) and π/6 lies in the principal range, the value is π/6, not 5π/6.
Traps
- Do not cancel inverse functions outside the principal branch.
- The range of tan inverse excludes ±π/2, so tan inverse(tan π/2) is undefined.
- In addition identities, check the product xy before using the formula.
- cos inverse(cos x) is not x for all x; reduce x to [0,π].
- Write principal values in radians unless the question specifies degrees.