Mathematics · Class 12 Science
Ch 2Inverse Trigonometric Functions — Class 12 Mathematics, concept-first.
Recall (Class XI) that a function has an inverse only when is bijective -- one-one (injective) and onto (surjective). If is bijective, its inverse is defined by and the domain of equals the range of , while the range of equals the domain of .
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Addition and Double-Angle Formulas for Inverse Trigonometric Functions
The formula follows from the tangent addition formula, but only holds directly when , which keeps inside the principal branch ; when and , the true sum exceeds and a correction of must be added instead: .
Most relevant Q&A
- Prove that $\tan^{-1}1+\tan^{-1}2+\tan^{-1}3=\pi$.Free
- Solve for $x$: $\tan^{-1}(2x)+\tan^{-1}(3x)=\dfrac{\pi}{4}$.Preview
- Simplify $\tan^{-1}\!\left(\dfrac12\right)+\tan^{-1}\!\left(\dfrac13\right)$.Preview
- Prove that $2\tan^{-1}\!\left(\dfrac13\right)=\tan^{-1}\!\left(\dfrac34\right)$.Preview
- If tan^-1 x + tan^-1 y + tan^-1 z = pi/2 and x + y + z = sqrt(3), then show that x = y = z.Preview
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Definition of Inverse Trigonometric Functions
Recall (Class XI) that a function has an inverse only when is bijective -- one-one (injective) and onto (surjective).
Principal Value Branches
Definition. Among all the intervals on which a given trigonometric function is one-one and onto its usual range, the principal value branch is the one interval singled out by convention -- chosen to c…
Domain and Range of Each Inverse Trigonometric Function
Because an inverse function's domain equals the range of the original function (restricted to its chosen branch), and its range equals that branch itself (Section 1), the domain and range of every inv…
Graphs of Inverse Trigonometric Functions
The reflection principle. If on a branch where is bijective, then lies on the graph of (restricted to that branch) exactly when lies on the graph of .
Elementary Properties and Identities
(A) Reciprocal identities. Since , if then , so and by the identical argument,
Summary
Definition and principal branch. A trigonometric function is not one-one on its full domain (periodicity), so its inverse is defined only after restricting to a branch on which it is bijective; the co…
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
+−Show 16 questionsHide questions16 questions
- Q1Solve: 2 sin^-1 x = cos^-1 x, 0 < x < 1.Preview
- Q2If tan^-1 x + tan^-1 y + tan^-1 z = pi/2 and x + y + z = sqrt(3), then show that x = y = z.Preview
- Q3The value of tan(pi/2 - tan^-1 (1/3)) is equal to (a) 1/3 (b) 3 (c) 2/3 (d) 3/2Preview
- Q4Find the value of sec²(cot⁻¹ 1/3) + cosec²(tan⁻¹ 1/2).Preview
- Q5The value of sec²(tan⁻¹2) + cosec²(cot⁻¹3) is (a) 12 (b) 13 (c) 15 (d) 5Preview
- Q6If sin(sin⁻¹(1/5) + cos⁻¹x) = 1, find x.Preview
- Q7Show that tan(π/4 + (1/2)cos⁻¹(a/b)) + tan(π/4 - (1/2)cos⁻¹(a/b)) = 2b/a.Preview
- Q8The principal value of sin^-1 sin(5π/6) is (a) π/6 (b) 5π/6 (c) π/2 (d) π/3Preview
- Q9If 0 < x < 1, then show that sin^-1 x + cos^-1 x = π/2.Preview
- Q10If cos^-1 x + cos^-1 y + cos^-1 z = π, then show that x² + y² + z² + 2xyz = 1.Preview
- Q11If tan⁻¹x + tan⁻¹y = 4π/5, then the value of cot⁻¹x + cot⁻¹y is (a) π (b) 3π/5 (c) 2π/5 (d) π/5Preview
- Q12If sin⁻¹x = tan⁻¹y, show that 1/x² - 1/y² = 1.Preview
- Q13Show that sin⁻¹(4/5) + sin⁻¹(5/13) + sin⁻¹(16/65) = π/2.Preview
- Q14If $\sin^{-1} x + \sin^{-1} y = \dfrac{2\pi}{3}$, then the value of $\cos^{-1} x + \cos^{-1} y$ is (a) $\dfrac{\pi}{3}$ (b) $\dfrac{\pi}{6}$…Preview
- Q15The value of $2\tan^{-1}\sqrt{x} - \cos^{-1}\left(\dfrac{1-x}{1+x}\right)$ is (a) $0$ (b) $1$ (c) $\dfrac{1}{3}$ (d) $\dfrac{1}{2}$Preview
- Q16The principal value of $\cot^{-1}\left(-\dfrac{1}{\sqrt{3}}\right)$ is (a) $\dfrac{2\pi}{3}$ (b) $-\dfrac{\pi}{3}$ (c) $\dfrac{\pi}{3}$ (d)…Preview
More questions
27 Q+−Show 3 questionsHide questions3 questions
+−Show 7 questionsHide questions7 questions
- Example 1Find the principal value of $\sin^{-1}\!\left(-\dfrac12\right)$.Free
- Example 2Find the principal value of $\cos^{-1}\!\left(-\dfrac{\sqrt3}{2}\right)$.Free
- Example 3Find the principal value of $\tan^{-1}(-1)$.Free
- Example 4Find the principal value of $\sec^{-1}(2)$.Preview
- Example 5Find the principal value of $\text{cosec}^{-1}(-\sqrt2)$.Preview
- Example 6Evaluate $\sin^{-1}\!\left(\sin\dfrac{2\pi}{3}\right)$.Preview
- Example 7Prove that $\sin^{-1}x+\cos^{-1}x=\dfrac{\pi}{2}$ for every $x\in[-1,1]$, and verify the result numerically for $x=\dfrac12$.Preview
+−Show 5 questionsHide questions5 questions
+−Show 4 questionsHide questions4 questions
+−Show 3 questionsHide questions3 questions
- Q17Describe the graph of $y=\tan^{-1}x$, stating its domain, range and behaviour as $x\to\pm\infty$.Free
- Q18Using the graph of $y=\sin^{-1}x$, state whether the function is increasing or decreasing, and find its value at $x=0$.Preview
- Q19Explain how the graph of $y=\cos^{-1}x$ is obtained from the graph of $y=\cos x$ restricted to $[0,\pi]$.Preview
+−Show 5 questionsHide questions5 questions
- Q20Prove that $\tan^{-1}x+\cot^{-1}x=\dfrac{\pi}{2}$ for every $x\in\mathbb{R}$.Free
- Q21Prove that $\cos^{-1}(-x)=\pi-\cos^{-1}x$ for $x\in[-1,1]$, and use it to find $\cos^{-1}\!\left(-\dfrac12\right)$.Free
- Q22Simplify $\tan^{-1}\!\left(\dfrac12\right)+\tan^{-1}\!\left(\dfrac13\right)$.Preview
- Q23Prove that $2\tan^{-1}\!\left(\dfrac13\right)=\tan^{-1}\!\left(\dfrac34\right)$.Preview
- Q24Show that $\sec^{-1}x+\text{cosec}^{-1}x=\dfrac{\pi}{2}$ for $x\ge1$, and verify the result numerically for $x=2$.Preview