Mathematics · Class 12 Science
Ch 4Inverse Trigonometric Functions — Class 12 Mathematics, concept-first.
Indirect measurement — finding a length or an angle without physically measuring it — is one of the oldest applications of trigonometry, and inverse trigonometric functions are the tool that makes it possible whenever the UNKNOWN quantity is an angle rather than a side.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Inverse Sine, Cosine and Tangent Functions — Domains and Ranges
Sine, cosine and tangent all fail to be one-to-one over their full domains, so each is first restricted to an interval on which it is a bijection, and the inverse is defined only on that restricted piece.
Most relevant Q&A
- Find all the values of $x$ such that (i) $-10\pi \le x \le 10\pi$ and $\sin x = 0$ (ii) $-3\pi \le x \le 3\pi$ and $\sin x = -1$.Free
- Find the period and amplitude of (i) $y = \sin 7x$ (ii) $y = -\sin\left(\dfrac13 x\right)$ (iii) $y = 4\sin(-2x)$.Free
- Sketch the graph of $y = \sin\left(\dfrac13 x\right)$ for $0 \le x < 6\pi$.Free
- Find the value of (i) $\sin^{-1}\left(\sin\dfrac{2\pi}3\right)$ (ii) $\sin^{-1}\left(\sin\dfrac{5\pi}4\right)$.Preview
- For what value of $x$ does $\sin x = \sin^{-1}x$?Preview
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Introduction
Indirect measurement — finding a length or an angle without physically measuring it — is one of the oldest applications of trigonometry, and inverse trigonometric functions are the tool that makes it…
Some Fundamental Concepts
Before any trigonometric function can be inverted, four pieces of background must be pinned down precisely: what makes a function periodic, what makes it odd or even, exactly what its domain and range…
Domain and Range of Trigonometric Functions
For a periodic real-valued function , a number is a period if lies in the domain of whenever does, and for every such ; the smallest such is called the period of .
Graphs of Functions
For , the graph of is the set of ordered pairs plotted in the -plane. Not every curve in the plane is the graph of a function: a curve represents a function exactly when it passes the vertical line te…
Amplitude and Period of a Graph
For a periodic graph, the amplitude is the maximum distance of the curve from the -axis — equivalently, the height from the axis up to a peak (or down to a trough).
Inverse Functions
A function is a rule that always returns a UNIQUE output for a given input. For a function to be invertible — for the reverse mapping (output back to input) to itself be a function — the original func…
Graphs of Inverse Functions
If is bijective (one-to-one and onto) with inverse , then exactly when . So the point lies on the graph of if and only if the point lies on the graph of — every point's coordinates are simply swapped.
Sine Function and Inverse Sine Function
Sine has domain and range ; its inverse is written or . This section builds the graph of , restricts it to a one-to-one piece, and defines the inverse sine function on that restricted piece.
+−Exercise 4.1i7 questions
- Q1Find all the values of $x$ such that (i) $-10\pi \le x \le 10\pi$ and $\sin x = 0$ (ii) $-3\pi \le x \le 3\pi$ and $\sin x = -1$.Free
- Q2Find the period and amplitude of (i) $y = \sin 7x$ (ii) $y = -\sin\left(\dfrac13 x\right)$ (iii) $y = 4\sin(-2x)$.Free
- Q3Sketch the graph of $y = \sin\left(\dfrac13 x\right)$ for $0 \le x < 6\pi$.Free
- Q4Find the value of (i) $\sin^{-1}\left(\sin\dfrac{2\pi}3\right)$ (ii) $\sin^{-1}\left(\sin\dfrac{5\pi}4\right)$.Preview
- Q5For what value of $x$ does $\sin x = \sin^{-1}x$?Preview
- Q6Find the domain of the following (i) $f(x) = \sin^{-1}\left(\dfrac{x^2+1}{2x}\right)$ (ii) $g(x) = 2\sin^{-1}(2x-1) - \dfrac{\pi}4$.Preview
- Q7Find the value of $\sin^{-1}\left(\sin\dfrac{5\pi}9\cos\dfrac{\pi}9 + \cos\dfrac{5\pi}9\sin\dfrac{\pi}9\right)$.Preview
The Graph of Sine Function
Since for every real , but for any smaller positive , the period of is exactly . So the graph over ANY interval of length — say — is repeated identically over every other interval of that length, e.g.
Properties of the Sine Function
Reading directly off the graph of :
The Inverse Sine Function and its Properties
Sine fails to be one-to-one over : every horizontal line , , crosses the sine curve infinitely many times, so sine does not pass the horizontal line test.
Graph of the Inverse Sine Function
receives and returns . As increases from to , increases from to ; connecting the plotted points with a smooth curve produces Fig. 4.6.
The Cosine Function and Inverse Cosine Function
Cosine has domain and range ; its inverse is written or . Since for all , but no smaller positive works for all , cosine's period is — the same period as sine, but the shape and the restricted domain…
+−Exercise 4.2i8 questions
- Q1Find all values of $x$ such that (i) $-6\pi \le x \le 6\pi$ and $\cos x = 0$ (ii) $-5\pi \le x \le 5\pi$ and $\cos x = 1$.Free
- Q2State the reason for $\cos^{-1}\left[\cos\left(-\dfrac{\pi}6\right)\right] \ne -\dfrac{\pi}6$.Free
- Q3Is $\cos^{-1}(-x) = \pi - \cos^{-1}(x)$ true? Justify your answer.Free
- Q4Find the principal value of $\cos^{-1}\left(\dfrac12\right)$.Preview
- Q5Find the value of (i) $2\cos^{-1}\left(\dfrac12\right) + \sin^{-1}\left(\dfrac12\right)$ (ii) $\cos^{-1}\left(\dfrac12\right) + \sin^{-1}(-1…Preview
- Q6Find the domain of (i) $f(x) = \sin^{-1}\left(\dfrac{|x|-2}3\right) + \cos^{-1}\left(\dfrac{1-|x|}4\right)$ (ii) $g(x) = \sin^{-1}x + \cos^{…Preview
- Q7For what value of $x$, the inequality $\dfrac{\pi}2 < \cos^{-1}(3x-1) < \pi$ holds?Preview
- Q8Find the value of (i) $\cos\left(\cos^{-1}\left(\dfrac45\right) + \sin^{-1}\left(\dfrac45\right)\right)$ (ii) $\cos^{-1}\left(\cos\dfrac{4\p…Preview
Graph of Cosine Function
Table of values on :
Properties of the Cosine Function
From the graph: cosine is continuous everywhere (no breaks), even (symmetric about the -axis), and for all ; the maximum occurs at , the minimum at
The Inverse Cosine Function and its Properties
Cosine is not one-to-one over , but restricting it to makes it one-to-one, still with range .
Graph of the Inverse Cosine Function
receives and returns . As increases from to , DECREASES from to — inverse cosine is a strictly DECREASING, continuous curve, shown in Fig. 4.14.
The Tangent Function and the Inverse Tangent Function
Tangent, , is used to find heights and distances (a building, mountain, or flagpole) from a known angle and a known side, or — inverted — an unknown angle from two known sides.
+−Exercise 4.3i4 questions
- Q1Find the domain of the following functions: (i) $\tan^{-1}\left(\sqrt{9-x^2}\right)$ (ii) $\dfrac12\tan^{-1}(1-x^2) - \dfrac{\pi}4$.Free
- Q2Find the value of (i) $\tan^{-1}\left(\tan\dfrac{5\pi}4\right)$ (ii) $\tan^{-1}\left(\tan\left(-\dfrac{\pi}6\right)\right)$.Free
- Q3Find the value of (i) $\tan\left(\tan^{-1}\left(\dfrac{7\pi}4\right)\right)$ (ii) $\tan\left(\tan^{-1}(1947)\right)$ (iii) $\tan\left(\tan^{…Preview
- Q4Find the value of (i) $\tan\left(\cos^{-1}\left(\dfrac12\right) - \sin^{-1}\left(-\dfrac12\right)\right)$ (ii) $\sin\left(\tan^{-1}\left(\df…Preview
The Graph of Tangent Function
Because the graph of is useful for reading off values over its repeated period, and tangent is ODD (so its graph is symmetric about the origin) with the SHORT period , it suffices to work out its shap…
Properties of the Tangent Function
From the graph:
The Inverse Tangent Function and its Properties
Tangent is not one-to-one over its full domain , but IS a bijection.
Graph of the Inverse Tangent Function
has domain the entire real line and range . Since is undefined (has vertical asymptotes) at , the graph of lies STRICTLY between the two horizontal lines and , approaching but never touching them — th…
The Cosecant Function and the Inverse Cosecant Function
is, like sine, an ODD function with period — its values repeat over every interval of length . Since is undefined wherever , its domain is .
Graph of the Cosecant Function
On , cosecant is continuous everywhere EXCEPT at (where ), and it has neither a maximum nor a minimum.
The Inverse Cosecant Function
Cosecant restricted to is a bijection onto its full range .
Graph of the Inverse Cosecant Function
has domain and range — that is, .
The Secant Function and Inverse Secant Function
is defined wherever , so its domain is . Since , never lands strictly between and , so its range is — the same range as cosecant.
The Graph of the Secant Function
On : in Quadrants I and IV (i.e. ), is positive; in Quadrants II and III (), is negative. Piece by piece: rises from to over ; rises from to over ; falls from to over ; and falls from to over .
Inverse Secant Function
Secant restricted to is a bijection onto its full range .
Graph of the Inverse Secant Function
has domain and range — that is, .
The Cotangent Function and the Inverse Cotangent Function
is defined for all except where or is undefined, i.e. except , — so its domain is and its range is . Like tangent, cotangent is an odd function with the shorter period .
The Graph of the Cotangent Function
Cotangent is continuous on . In Quadrants I and III it is positive; in Quadrants II and IV it is negative.
Inverse Cotangent Function
Cotangent is not one-to-one over its full domain, but IS a bijection.
Graph of the Inverse Cotangent Function
has domain and range — that is, , the only one of the six inverse trig functions defined for EVERY real number with no gap in either domain or output.
Principal Value of Inverse Trigonometric Functions
The principal value of an inverse trigonometric function at a point is the value of that inverse function at that lies inside its principal-value branch (range). E.g.
+−Exercise 4.4i2 questions
Properties of Inverse Trigonometric Functions
30 QThis section develops the ten families of working properties of inverse trigonometric functions — valid strictly WITHIN each function's principal value branch, and only where each side is defined.
+−Exercise 4.5i10 questions
- Q1Find the value, if it exists. If not, give the reason for non-existence. (i) $\sin^{-1}(\cos \pi)$ (ii) $\tan^{-1}\left(\sin\left(-\dfrac{5\…Free
- Q2Find the value of the expression in terms of $x$, with the help of a reference triangle. (i) $\sin\left(\cos^{-1}(1-x)\right)$ (ii) $\cos\le…Free
- Q3Find the value of (i) $\sin^{-1}\left(\cos\left(\sin^{-1}\dfrac{\sqrt3}2\right)\right)$ (ii) $\cot\left(\sin^{-1}\dfrac35 + \sin^{-1}\dfrac4…Free
- Q4Prove that (i) $\tan^{-1}\dfrac2{11} + \tan^{-1}\dfrac7{24} = \tan^{-1}\dfrac12$ (ii) $\sin^{-1}\dfrac35 - \cos^{-1}\dfrac{12}{13} = \sin^{-…Preview
- Q5Prove that $\tan^{-1}x + \tan^{-1}y + \tan^{-1}z = \tan^{-1}\left[\dfrac{x+y+z-xyz}{1-xy-yz-zx}\right]$.Preview
- Q6If $\tan^{-1}x + \tan^{-1}y + \tan^{-1}z = \pi$, show that $x+y+z = xyz$.Preview
- Q7Prove that $\tan^{-1}x + \tan^{-1}\dfrac{2x}{1-x^2} = \tan^{-1}\dfrac{3x-x^3}{1-3x^2}$, $|x| < \dfrac1{\sqrt3}$.Preview
- Q8Simplify: $\tan^{-1}\dfrac{x}y - \tan^{-1}\dfrac{x-y}{x+y}$.Preview
- Q9Solve: (i) $\sin^{-1}\dfrac5x + \sin^{-1}\dfrac{12}x = \dfrac{\pi}2$ (ii) $2\tan^{-1}x = \cos^{-1}\dfrac{1-a^2}{1+a^2} - \cos^{-1}\dfrac{1-b…Preview
- Q10Find the number of solutions of the equation $\tan^{-1}(x-1) + \tan^{-1}x + \tan^{-1}(x+1) = \tan^{-1}(3x)$.Preview
+−Exercise 4.6i20 questions
- Q1The value of $\sin^{-1}(\cos x)$, $0 \le x \le \pi$ is (1) $\pi - x$ (2) $x - \dfrac{\pi}2$ (3) $\dfrac{\pi}2 - x$ (4) $x - \pi$Free
- Q2If $\sin^{-1}x + \sin^{-1}y = \dfrac{2\pi}3$; then $\cos^{-1}x + \cos^{-1}y$ is equal to (1) $\dfrac{2\pi}3$ (2) $\dfrac{\pi}3$ (3) $\dfrac{…Free
- Q3$\sin^{-1}\dfrac35 - \cos^{-1}\dfrac{12}{13} + \sec^{-1}\dfrac53 - \text{cosec}^{-1}\dfrac{13}{12}$ is equal to (1) $2\pi$ (2) $\pi$ (3) $0$…Free
- Q4If $\sin^{-1}x = 2\sin^{-1}\alpha$ has a solution, then (1) $|\alpha| \le \dfrac1{\sqrt2}$ (2) $|\alpha| \ge \dfrac1{\sqrt2}$ (3) $|\alpha|…Preview
- Q5$\sin^{-1}(\cos x) = \dfrac{\pi}2 - x$ is valid for (1) $-\pi \le x \le 0$ (2) $0 \le x \le \pi$ (3) $-\dfrac{\pi}2 \le x \le \dfrac{\pi}2$…Preview
- Q6If $\sin^{-1}x + \sin^{-1}y + \sin^{-1}z = \dfrac{3\pi}2$, the value of $x^{2017}+y^{2018}+z^{2019} - \dfrac9{x^{101}+y^{101}+z^{101}}$ is (…Preview
- Q7If $\cot^{-1}x = \dfrac{2\pi}5$ for some $x \in \mathbb{R}$, the value of $\tan^{-1}x$ is (1) $-\dfrac{\pi}{10}$ (2) $\dfrac{\pi}5$ (3) $\df…Preview
- Q8The domain of the function defined by $f(x) = \sin^{-1}\sqrt{x-1}$ is (1) $[1,\,2]$ (2) $[-1,\,1]$ (3) $[0,\,1]$ (4) $[-1,\,0]$Preview
- Q9If $x = \dfrac15$, the value of $\cos\left(\cos^{-1}x + 2\sin^{-1}x\right)$ is (1) $-\sqrt{\dfrac{24}{25}}$ (2) $\sqrt{\dfrac{24}{25}}$ (3)…Preview
- Q10$\tan^{-1}\left(\dfrac14\right) + \tan^{-1}\left(\dfrac29\right)$ is equal to (1) $\dfrac12\cos^{-1}\left(\dfrac35\right)$ (2) $\dfrac12\sin…Preview
- Q11If the function $f(x) = \sin^{-1}(x^2-3)$, then $x$ belongs to (1) $[-1,\,1]$ (2) $[\sqrt2,\,2]$ (3) $[-2,\,-\sqrt2]\cup[\sqrt2,\,2]$ (4) $[…Preview
- Q12If $\cot^{-1}2$ and $\cot^{-1}3$ are two angles of a triangle, then the third angle is (1) $\dfrac{\pi}4$ (2) $\dfrac{3\pi}4$ (3) $\dfrac{\p…Preview
- Q13$\sin^{-1}\left(\tan\dfrac{\pi}4\right) - \sin^{-1}\left(\sqrt{\dfrac3x}\right) = \dfrac{\pi}6$. Then $x$ is a root of the equation (1) $x^2…Preview
- Q14$\sin^{-1}\left(2\cos^2x - 1\right) + \cos^{-1}\left(1-2\sin^2x\right) =$ (1) $\dfrac{\pi}2$ (2) $\dfrac{\pi}3$ (3) $\dfrac{\pi}4$ (4) $\dfr…Preview
- Q15If $\cot^{-1}\left(\sqrt{\sin\alpha}\right) + \tan^{-1}\left(\sqrt{\sin\alpha}\right) = u$, then $\cos 2u$ is equal to (1) $\tan^2\alpha$ (2…Preview
- Q16If $|x| \le 1$, then $2\tan^{-1}x - \sin^{-1}\dfrac{2x}{1+x^2}$ is equal to (1) $\tan^{-1}x$ (2) $\sin^{-1}x$ (3) $0$ (4) $\pi$Preview
- Q17The equation $\tan^{-1}x - \cot^{-1}x = \tan^{-1}\left(\dfrac1{\sqrt3}\right)$ has (1) no solution (2) unique solution (3) two solutions (4)…Preview
- Q18If $\sin^{-1}x + \cot^{-1}\left(\dfrac12\right) = \dfrac{\pi}2$, then $x$ is equal to (1) $\dfrac12$ (2) $\dfrac1{\sqrt5}$ (3) $\dfrac2{\sqr…Preview
- Q19If $\sin^{-1}\dfrac{x}5 + \text{cosec}^{-1}\dfrac54 = \dfrac{\pi}2$, then the value of $x$ is (1) $4$ (2) $5$ (3) $2$ (4) $3$Preview
- Q20$\sin(\tan^{-1}x)$, $|x| < 1$ is equal to (1) $\dfrac{x}{\sqrt{1-x^2}}$ (2) $\dfrac1{\sqrt{1-x^2}}$ (3) $\dfrac1{\sqrt{1+x^2}}$ (4) $\dfrac{…Preview
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
+−Show 20 questionsHide questions20 questions
- Q1Prove that $\tan^{-1}x < x$, for all $x>0$.Preview
- Q2If $\sin^{-1}x + \sin^{-1}y = \dfrac{2\pi}{3}$, then $\cos^{-1}x + \cos^{-1}y$ is equal to : (a) $\pi$ (b) $\dfrac{2\pi}{3}$ (c) $\dfrac{\pi…Preview
- Q3$\tan^{-1}\left(\dfrac{1}{4}\right) + \tan^{-1}\left(\dfrac{2}{9}\right)$ is : (a) $\tan^{-1}\left(\dfrac{1}{2}\right)$ (b) $\dfrac{1}{2}\co…Preview
- Q4Find the value of $\sin^{-1}\left[\sin\left(\dfrac{5\pi}{4}\right)\right]$.Preview
- Q5(a) Draw the graph of $\cos x$ in $[0, \pi]$ and $\cos^{-1}x$ in $[-1, 1]$. **OR** (b) Find the equation of the circle passing through the p…Preview
- Q6The principal value of $\cos^{-1}\left(\dfrac{\sqrt3}{2}\right)$ is : (a) $\dfrac{\pi}{2}$ (b) $\dfrac{\pi}{3}$ (c) $\dfrac{5\pi}{6}$ (d) $\…Preview
- Q7Find the principal value of $\tan^{-1}(\sqrt3)$.Preview
- Q8If $3\cos^{-1}x=\cos^{-1}(4x^3-3x)$, (a) $x\in\left(\dfrac12, 1\right)$ (b) $x\in\left[\dfrac12, 1\right]$ (c) $x\in(-\infty, 1]$ (d) $x\in\…Preview
- Q9The number of real numbers in $[0, 2\pi]$ satisfying $\sin^4x-2\sin^2x+1$ is : (a) $1$ (b) $2$ (c) $\infty$ (d) $4$Preview
- Q10The Principal value of $\sin^{-1}\left(\dfrac{-1}{2}\right)$ is : (a) $\dfrac{-\pi}{6}$ (b) $0$ (c) $\dfrac{-\pi}{2}$ (d) $\dfrac{\pi}{2}$Preview
- Q11For what value of x, the inequality $\dfrac{\pi}{2}<\cos^{-1}(3x-1)<\pi$ holds ?Preview
- Q12If $\sin^{-1}x+\cot^{-1}\left(\dfrac12\right)=\dfrac{\pi}{2}$, then x is equal to : (a) $\dfrac{2}{\sqrt5}$ (b) $\dfrac12$ (c) $\dfrac{\sqrt…Preview
- Q13Show that $\cot^{-1}\left(\dfrac{1}{\sqrt{x^2-1}}\right)=\sec^{-1}x$, $|x|>1$.Preview
- Q14The domain of the function defined by $f(x)=\sin^{-1}\sqrt{x-1}$ is : (a) $[0, 1]$ (b) $[1, 2]$ (c) $[-1, 0]$ (d) $[-1, 1]$Preview
- Q15The number of real numbers in $[0, 2\pi]$ satisfying $\sin^4x-2\sin^2x+1$ is : (a) $1$ (b) $2$ (c) $\infty$ (d) $4$Preview
- Q16Find the value of $\tan^{-1}(-\sqrt3)$.Preview
- Q17Prove that $\tan^{-1}\dfrac{2}{11}+\tan^{-1}\dfrac{7}{24}=\tan^{-1}\dfrac12$Preview
- Q18If $x<0$, then $\tan^{-1}\left(\dfrac1x\right)$ is equal to : (a) $-\pi+\cot^{-1}(x)$ (b) $\tan^{-1}(x)$ (c) $-\pi+\tan^{-1}x$ (d) $\cot^{-1…Preview
- Q19Simplify : $\sin^{-1}[\sin10]$Preview
- Q20Prove that $2\tan^{-1}\dfrac12+\tan^{-1}\dfrac17=\tan^{-1}\dfrac{31}{17}$Preview