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Inverse Trigonometric Functions formulas for JEE

Every Inverse Trigonometric Functions formula you need for JEE, grouped by concept.

33 formulas2 concepts
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QuestionState the formula — Inverse Sine of Sine
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Inverse Sine of Sinesin⁡−1(sin⁡x)=x\sin^{-1}(\sin x) = xApplying the inverse sine function to sine yields the original angle within the principal branch.applies whenx∈[−π2,π2]x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]trigonometryinversebasics
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All 33 Inverse Trigonometric Functions formulas
01

Inverse Trig Basics & Graphs

2 formulas

Inverse Sine of Sine

sin⁡−1(sin⁡x)=x\sin^{-1}(\sin x) = x

Applying the inverse sine function to sine yields the original angle within the principal branch.

applies whenx∈[−π2,π2]x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]
trigonometryinversebasics

Sine of Inverse Sine

sin⁡(sin⁡−1x)=x\sin(\sin^{-1} x) = x

Applying the sine function to its inverse yields the original argument.

applies whenx∈[−1,1]x \in [-1, 1]
trigonometryinversebasics
02

Properties of Inverse Trigonometric Functions

31 formulas

Double Inverse Tangent to Sine

sin⁡−1(2x1+x2)=2tan⁡−1x\sin^{-1}\left(\frac{2x}{1+x^2}\right) = 2\tan^{-1} x

Conversion of double inverse tangent into an inverse sine function.

applies when∣x∣≤1|x| \le 1
trigonometryinversemultiplejee-advanced

Double Inverse Tangent to Cosine

cos⁡−1(1−x21+x2)=2tan⁡−1x\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) = 2\tan^{-1} x

Conversion of double inverse tangent into an inverse cosine function.

applies whenx≥0x \ge 0
trigonometryinversemultiplejee-advanced

Double Inverse Tangent to Tangent

tan⁡−1(2x1−x2)=2tan⁡−1x\tan^{-1}\left(\frac{2x}{1-x^2}\right) = 2\tan^{-1} x

Conversion of double inverse tangent into a single inverse tangent function.

applies when∣x∣<1|x| < 1
trigonometryinversemultiplejee-advanced

Complementary Inverse Secant Cosecant

sec⁡−1x+cosec−1x=π2\sec^{-1} x + \text{cosec}^{-1} x = \frac{\pi}{2}

The sum of complementary inverse secant and cosecant is pi/2.

applies when∣x∣≥1|x| \ge 1
trigonometryinversecomplementaryjee-advanced

Complementary Inverse Tangent Cotangent

tan⁡−1x+cot⁡−1x=π2\tan^{-1} x + \cot^{-1} x = \frac{\pi}{2}

The sum of complementary inverse tangent and cotangent is pi/2.

applies whenx∈Rx \in \mathbb{R}
trigonometryinversecomplementaryjee-advanced

Inverse Cosine Addition

cos⁡−1x+cos⁡−1y=cos⁡−1(xy−1−x21−y2)\cos^{-1} x + \cos^{-1} y = \cos^{-1}\left(xy - \sqrt{1-x^2}\sqrt{1-y^2}\right)

Addition formula for two inverse cosine functions.

applies whenx,y≥0x, y \ge 0
trigonometryinverseadditionjee-advanced

Inverse Cotangent Rationalization

cot⁡−1(1+sin⁡x+1−sin⁡x1+sin⁡x−1−sin⁡x)=x2\cot^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right) = \frac{x}{2}

Simplification using rationalization of root functions of sine.

applies whenx∈(0,π4)x \in \left(0, \frac{\pi}{4}\right)
trigonometryinversesimplification

Double Inverse Cosine

sin⁡−1(2x1−x2)=2cos⁡−1x\sin^{-1}(2x\sqrt{1-x^2}) = 2\cos^{-1} x

Simplification identity for an algebraic argument reducing to double inverse cosine.

applies when12≤x≤1\frac{1}{\sqrt{2}} \le x \le 1
trigonometryinversesimplification

Double Inverse Sine

sin⁡−1(2x1−x2)=2sin⁡−1x\sin^{-1}(2x\sqrt{1-x^2}) = 2\sin^{-1} x

Simplification identity for an algebraic argument reducing to double inverse sine.

applies when−12≤x≤12\frac{-1}{\sqrt{2}} \le x \le \frac{1}{\sqrt{2}}
trigonometryinversesimplification

Negative Argument Inverse Cosine

cos⁡−1(−x)=π−cos⁡−1x\cos^{-1}(-x) = \pi - \cos^{-1} x

Property of inverse cosine with a negative argument.

applies whenx∈[−1,1]x \in [-1, 1]
trigonometryinversenegativejee-advanced

Negative Argument Inverse Sine

sin⁡−1(−x)=−sin⁡−1x\sin^{-1}(-x) = -\sin^{-1} x

Property of inverse sine with a negative argument (Odd function analogue).

applies whenx∈[−1,1]x \in [-1, 1]
trigonometryinversenegativejee-advanced

Negative Argument Inverse Tangent

tan⁡−1(−x)=−tan⁡−1x\tan^{-1}(-x) = -\tan^{-1} x

Property of inverse tangent with a negative argument (Odd function analogue).

applies whenx∈Rx \in \mathbb{R}
trigonometryinversenegativejee-advanced

Reciprocal Inverse Cosine

cos⁡−1(1x)=sec⁡−1x\cos^{-1}\left(\frac{1}{x}\right) = \sec^{-1} x

Inverse cosine of reciprocal mapping to inverse secant.

applies when∣x∣≥1|x| \ge 1
trigonometryinversereciprocaljee-advanced

Reciprocal Inverse Sine

sin⁡−1(1x)=cosec−1x\sin^{-1}\left(\frac{1}{x}\right) = \text{cosec}^{-1} x

Inverse sine of reciprocal mapping to inverse cosecant.

applies when∣x∣≥1|x| \ge 1
trigonometryinversereciprocaljee-advanced

Reciprocal Inverse Tangent (Positive)

tan⁡−1(1x)=cot⁡−1x\tan^{-1}\left(\frac{1}{x}\right) = \cot^{-1} x

Inverse tangent of reciprocal mapping to inverse cotangent for positive x.

applies whenx>0x > 0
trigonometryinversereciprocaljee-advanced

Reciprocal Inverse Tangent (Negative)

tan⁡−1(1x)=−π+cot⁡−1x\tan^{-1}\left(\frac{1}{x}\right) = -\pi + \cot^{-1} x

Inverse tangent of reciprocal mapping to inverse cotangent for negative x.

applies whenx<0x < 0
trigonometryinversereciprocaljee-advanced

Inverse Cotangent to Secant

cot⁡−1(1x2−1)=sec⁡−1x\cot^{-1}\left(\frac{1}{\sqrt{x^2-1}}\right) = \sec^{-1} x

Simplification of inverse cotangent with an algebraic square root argument.

applies when∣x∣>1|x| > 1
trigonometryinversesimplification

Inverse Sine Addition

sin⁡−1x+sin⁡−1y=sin⁡−1(x1−y2+y1−x2)\sin^{-1} x + \sin^{-1} y = \sin^{-1}\left(x\sqrt{1-y^2} + y\sqrt{1-x^2}\right)

Addition formula for two inverse sine functions.

applies whenx,y≥0,x2+y2≤1x, y \ge 0, x^2 + y^2 \le 1
trigonometryinverseadditionjee-advanced

Inverse Tangent Subtraction

tan⁡−1x−tan⁡−1y=tan⁡−1(x−y1+xy)\tan^{-1} x - \tan^{-1} y = \tan^{-1}\left(\frac{x-y}{1+xy}\right)

Subtraction formula for inverse tangents yielding a single inverse tangent.

applies whenxy>−1xy > -1
trigonometryinversesubtractionjee-advanced

Inverse Tangent Simplification 1

tan⁡−1(cos⁡x1−sin⁡x)=π4+x2\tan^{-1}\left(\frac{\cos x}{1 - \sin x}\right) = \frac{\pi}{4} + \frac{x}{2}

Simplification of inverse tangent involving cosine and sine ratio.

applies when−3π2<x<π2-\frac{3\pi}{2} < x < \frac{\pi}{2}
trigonometryinversesimplification

Inverse Tangent Simplification 2

tan⁡−1(1+x2−1x)=12tan⁡−1x\tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right) = \frac{1}{2}\tan^{-1} x

Simplification requiring a standard tangent substitution.

applies whenx≠0x \neq 0
trigonometryinversesimplification

Inverse Tangent Simplification 3

tan⁡−1(1−cos⁡x1+cos⁡x)=x2\tan^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right) = \frac{x}{2}

Simplification using half-angle cosine identities.

applies when0<x<π0 < x < \pi
trigonometryinversesimplification

Inverse Tangent Simplification 4

tan⁡−1(cos⁡x−sin⁡xcos⁡x+sin⁡x)=π4−x\tan^{-1}\left(\frac{\cos x - \sin x}{\cos x + \sin x}\right) = \frac{\pi}{4} - x

Simplification by dividing numerator and denominator by cosine.

applies when−π4<x<3π4-\frac{\pi}{4} < x < \frac{3\pi}{4}
trigonometryinversesimplification

Inverse Tangent to Sine Conversion

tan⁡−1(xa2−x2)=sin⁡−1xa\tan^{-1}\left(\frac{x}{\sqrt{a^2-x^2}}\right) = \sin^{-1}\frac{x}{a}

Algebraic form in inverse tangent equivalent to inverse sine.

applies when∣x∣<a|x| < a
trigonometryinversesimplification

Inverse Tangent Rationalization

tan⁡−1(1+x−1−x1+x+1−x)=π4−12cos⁡−1x\tan^{-1}\left(\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right) = \frac{\pi}{4} - \frac{1}{2}\cos^{-1}x

Simplification using rationalization and substitution for algebraic argument.

applies when−12≤x≤1-\frac{1}{\sqrt{2}} \le x \le 1
trigonometryinversesimplification

Inverse Tangent Addition

tan⁡−1x+tan⁡−1y=tan⁡−1(x+y1−xy)\tan^{-1} x + \tan^{-1} y = \tan^{-1}\left(\frac{x+y}{1-xy}\right)

Addition formula for inverse tangents yielding a single inverse tangent.

applies whenxy<1xy < 1
trigonometryinverseadditionjee-advanced

Inverse Tangent Addition (Obtuse)

tan⁡−1x+tan⁡−1y=π+tan⁡−1(x+y1−xy)\tan^{-1} x + \tan^{-1} y = \pi + \tan^{-1}\left(\frac{x+y}{1-xy}\right)

Addition formula for inverse tangents when product is greater than 1.

applies whenx>0,y>0,xy>1x > 0, y > 0, xy > 1
trigonometryinverseadditionjee-advanced

Triple Inverse Cosine

cos⁡−1(4x3−3x)=3cos⁡−1x\cos^{-1}(4x^3 - 3x) = 3\cos^{-1} x

Polynomial argument in inverse cosine reducing to triple inverse cosine.

applies whenx∈[12,1]x \in \left[\frac{1}{2}, 1\right]
trigonometryinversesimplification

Triple Inverse Sine

sin⁡−1(3x−4x3)=3sin⁡−1x\sin^{-1}(3x - 4x^3) = 3\sin^{-1} x

Polynomial argument in inverse sine reducing to triple inverse sine.

applies whenx∈[−12,12]x \in \left[-\frac{1}{2}, \frac{1}{2}\right]
trigonometryinversesimplification

Triple Inverse Tangent

tan⁡−1(3x−x31−3x2)=3tan⁡−1x\tan^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right) = 3\tan^{-1} x

Rational algebraic argument in inverse tangent reducing to triple inverse tangent.

applies when−13<x<13-\frac{1}{\sqrt{3}} < x < \frac{1}{\sqrt{3}}
trigonometryinversesimplification

Complementary Inverse Sine Cosine

sin⁡−1x+cos⁡−1x=π2\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}

The sum of complementary inverse trigonometric functions is pi/2.

applies whenx∈[−1,1]x \in [-1, 1]
trigonometryinversecomplementaryjee-advanced
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