CBSE · Class 12 · Mathematics · Chapter 2

Inverse Trigonometric Functions — Formula Sheet

Board Formulas

18 formulas

1.Principal Branch of sin⁻¹★
$sin^{- 1} : [- 1, 1] \to [- \frac{π}{2}, \frac{π}{2}]$
Sine is one-one on [−π/2, π/2], and this interval is the range of sin⁻¹. Negative inputs give negative angles.
2.Principal Branch of cos⁻¹★
$cos^{- 1} : [- 1, 1] \to [0, π]$
The range is [0, π], so cos⁻¹ is never negative. Negative inputs give obtuse angles.
3.Principal Branch of tan⁻¹★
$tan^{- 1} : R \to (- \frac{π}{2}, \frac{π}{2})$
Defined for every real number. The end-points ±π/2 are never reached.
4.Principal Branch of cot⁻¹
$cot^{- 1} : R \to (0, π)$
Same interval as cos⁻¹, but open at both ends. cot⁻¹(−1) = 3π/4, not −π/4.
5.Principal Branch of sec⁻¹
$sec^{- 1} : R - (- 1, 1) \to [0, π] - {\frac{π}{2}}$
Defined only for |x| ≥ 1. π/2 is left out because sec π/2 is undefined.
6.Principal Branch of cosec⁻¹
$cosec^{- 1} : R - (- 1, 1) \to [- \frac{π}{2}, \frac{π}{2}] - {0}$
Defined only for |x| ≥ 1. 0 is left out because cosec 0 is undefined.
7.Meaning of an Inverse Trig Value
$y = sin^{- 1} x \Leftrightarrow sin y = x, y \in [- \frac{π}{2}, \frac{π}{2}]$
To find a principal value, find the angle in the branch whose sine (or cosine, ...) equals x. The graph of y = sin⁻¹x is the mirror image of the restricted sine graph in the line y = x.
8.Notation Warning
$sin^{- 1} x \neq = (sin x)^{- 1} = \frac{1}{s i n x}$
The −1 means inverse function, not reciprocal. The same applies to cos⁻¹, tan⁻¹ and the rest.
9.Function of its Inverse
$sin (sin^{- 1} x) = x, cos (cos^{- 1} x) = x (- 1 \leq x \leq 1); tan (tan^{- 1} x) = x (x \in R)$
Valid wherever the inverse is defined. sin(sin⁻¹ 2) has no meaning, because 2 is outside [−1, 1].
10.Inverse of the Function (sin, tan)★
$sin^{- 1} (sin x) = x (- \frac{π}{2} \leq x \leq \frac{π}{2}), tan^{- 1} (tan x) = x (- \frac{π}{2} < x < \frac{π}{2})$
Valid ONLY when x is already in the principal branch. This is the step behind every 'simplest form' question.
11.Inverse of the Function (cos, cot)
$cos^{- 1} (cos x) = x (0 \leq x \leq π), cot^{- 1} (cot x) = x (0 < x < π)$
Valid only on the branch [0, π] (open for cot). Outside it, shift the angle into the branch first.
12.sin⁻¹(sin x) Outside the Branch★
$sin^{- 1} (sin x) = π - x, \frac{π}{2} \leq x \leq \frac{3 π}{2}$
Uses sin x = sin(π − x), and π − x lies in [−π/2, π/2]. Example: sin⁻¹(sin 4π/5) = π/5.
13.cos⁻¹(cos x) Outside the Branch
$cos^{- 1} (cos x) = 2 π - x, π \leq x \leq 2 π$
Uses cos x = cos(2π − x), and 2π − x lies in [0, π]. Example: cos⁻¹(cos 5π/3) = π/3.
14.tan⁻¹(tan x) Outside the Branch
$tan^{- 1} (tan x) = x - π, \frac{π}{2} < x < \frac{3 π}{2}$
tan has period π, so subtract π to bring x into (−π/2, π/2). Example: tan⁻¹(tan 4π/3) = π/3.
15.Negative Arguments: sin⁻¹, tan⁻¹, cosec⁻¹
$sin^{- 1} (- x) = - sin^{- 1} x, tan^{- 1} (- x) = - tan^{- 1} x, cosec^{- 1} (- x) = - cosec^{- 1} x$
Valid for every x in the domain: −1 ≤ x ≤ 1 for sin⁻¹, all real x for tan⁻¹, |x| ≥ 1 for cosec⁻¹. These branches are symmetric about 0, so a negative input gives the negative of the angle. This follows from the branch table. In a board answer, show the step: 'sin y = −1/2 with y in [−π/2, π/2] gives y = −π/6'.
16.Negative Arguments: cos⁻¹, cot⁻¹, sec⁻¹★
$cos^{- 1} (- x) = π - cos^{- 1} x, cot^{- 1} (- x) = π - cot^{- 1} x, sec^{- 1} (- x) = π - sec^{- 1} x$
Valid for every x in the domain: −1 ≤ x ≤ 1 for cos⁻¹, all real x for cot⁻¹, |x| ≥ 1 for sec⁻¹. These branches lie in [0, π], so a negative input gives π minus the angle. NCERT uses the same step for cot⁻¹(−1/√3) = π − π/3 = 2π/3.
17.Domain Condition for sin⁻¹ and cos⁻¹
$sin^{- 1} (g (x)), cos^{- 1} (g (x)) defined \Leftrightarrow - 1 \leq g (x) \leq 1$
For sec⁻¹(g(x)) or cosec⁻¹(g(x)) the condition is |g(x)| ≥ 1. tan⁻¹ and cot⁻¹ accept every real value.
18.Substitutions for Simplest Form
$a^{2} - x^{2} : x = a sin θ; a^{2} + x^{2} : x = a tan θ; x^{2} - a^{2} : x = a sec θ$
After substituting, simplify with a trig identity and use f⁻¹(f(θ)) = θ. Check that θ (or the angle you end with) lies in the principal branch for the given range of x.