Inverse Trigonometric Functions
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Direct answer
Each inverse trigonometric function is defined on a restricted domain with a principal value range: sin^-1 takes [-1, 1] to [-pi/2, pi/2], cos^-1 takes [-1, 1] to [0, pi], and tan^-1 takes all reals to (-pi/2, pi/2). The complement identities sin^-1 x + cos^-1 x = pi/2 on [-1, 1] and tan^-1 x + cot^-1 x = pi/2 are everyday tools, and every sum formula carries conditions that must be checked before use.
What you must remember
- Principal ranges: sin^-1 x in [-pi/2, pi/2] for x in [-1, 1]; cos^-1 x in [0, pi]; tan^-1 x in (-pi/2, pi/2); cot^-1 x in (0, pi); sec^-1 and cosec^-1 need |x| >= 1.
- Complement identities: sin^-1 x + cos^-1 x = pi/2; tan^-1 x + cot^-1 x = pi/2; sec^-1 x + cosec^-1 x = pi/2 — each valid on its full domain.
- Tangent sums: tan^-1 x + tan^-1 y = tan^-1((x + y)/(1 - xy)) when xy < 1; add pi when xy > 1 with x, y > 0; subtract pi when xy > 1 with x, y < 0; the difference form tan^-1 x - tan^-1 y = tan^-1((x - y)/(1 + xy)) holds when xy > -1.
- Doubling: 2 tan^-1 x = sin^-1(2x/(1 + x^2)) = cos^-1((1 - x^2)/(1 + x^2)) for |x| <= 1.
- Sine sums: sin^-1 x + sin^-1 y = sin^-1(x sqrt(1 - y^2) + y sqrt(1 - x^2)) when x^2 + y^2 <= 1 (with pi-adjustment otherwise); cos^-1 x + cos^-1 y follows the same pattern with the cross product form.
- The trap pair: sin^-1(sin x) = x only for x in [-pi/2, pi/2]; cos^-1(cos x) = x only for x in [0, pi]; tan^-1(tan x) = x only for x in (-pi/2, pi/2).
- Reflection shift: tan^-1(1/x) = pi/2 - tan^-1 x for x > 0, and -pi/2 - tan^-1 x for x < 0.
Common confusion
Applying sin^-1(sin x) = x outside the principal range is the most reliable trap in the chapter — the answer must be the equivalent angle inside the range, so sin^-1(sin(2pi/3)) is pi/3, not 2pi/3. The twin error is ignoring the xy < 1 condition on tangent sums: tan^-1 2 + tan^-1 3 equals 3pi/4, not the -pi/4 that blind substitution returns. Conditions are not decoration here; they are the question.
Exam-focused takeaway
JEE Main asks principal values, complement identities, conditional tangent sums and sin^-1(sin x) adjustments — statement and numerical questions where the conditions do the filtering. JEE Advanced builds multi-step simplifications where each step's condition must be verified, equations set in inverse trigonometric notation, and expressions fused with calculus. The discipline: before each identity, confirm the arguments lie where the formula is honest.
Frequently asked questions
What is sin^-1(sin x) for x outside [-pi/2, pi/2]?
The equivalent angle within the principal range whose sine matches — for example sin^-1(sin(2pi/3)) = pi/3; never the raw x.
When does tan^-1 x + tan^-1 y need a pi adjustment?
When xy > 1: add pi if x, y > 0 and subtract pi if x, y < 0; otherwise the unadjusted tangent formula holds.
What is sin^-1 x + cos^-1 x?
pi/2 for every x in [-1, 1] — the pair splits the right angle, just as tan^-1 x + cot^-1 x = pi/2 for all real x.
What is 2 tan^-1 x equal to?
sin^-1(2x/(1 + x^2)) and cos^-1((1 - x^2)/(1 + x^2)) for |x| <= 1; outside that band the equivalent-angle adjustments apply.
Does sin^-1 2 exist?
No. The domain of sin^-1 is [-1, 1]; any argument outside it — including from intermediate algebra — means the expression is undefined, a favourite statement-question filter.