If \(a=sin^{-1}(sin\ 5)\) and \(b=cos^{-1}(cos\ 5)\), then \(a^2+b^2=\)
Solution and Explanation
The answer is \(8\pi^2-40\pi+50\).
Top Questions on Inverse Trigonometric Functions
- If the domain of the function \(f(x) = \cos^{-1}\left(\frac{2x-5}{11-3x}\right) + \sin^{-1}(2x^2-3x+1)\) is the interval \([\alpha, \beta]\), then \(\alpha + 2\beta\) is equal to:}
- JEE Main - 2026
- Mathematics
- Inverse Trigonometric Functions
- The number of solutions of \[ \tan^{-1}(4x) + \tan^{-1}(6x) = \frac{\pi}{6}, \] where \[ -\frac{1}{2\sqrt{6}}<x<\frac{1}{2\sqrt{6}}, \] is equal to
- JEE Main - 2026
- Mathematics
- Inverse Trigonometric Functions
- If the domain of the function $f(x)=\sin^{-1}\!\left(\dfrac{1}{x^2-2x-2}\right)$ is $(-\infty,\alpha)\cup[\beta,\gamma]\cup[\delta,\infty)$, then $\alpha+\beta+\gamma+\delta$ is equal to
- JEE Main - 2026
- Mathematics
- Inverse Trigonometric Functions
- If domain of $f(x)=\sin^{-1}\!\left(\dfrac{1}{x^2-2x-2}\right)$ is $(-\infty,\alpha]\cup[\beta,\gamma]\cup[\delta,\infty)$, then $(\alpha+\beta+\gamma+\delta)$ is
- JEE Main - 2026
- Mathematics
- Inverse Trigonometric Functions
- Let the maximum value of \( (\sin^{-1}x)^2 + (\cos^{-1}x)^2 \) for \( x \in \left[ -\frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}} \right] \) be \( \frac{m}{n}\pi^2 \), where \( \gcd(m, n) = 1 \). Then \( m + n \) is equal to _________.
- JEE Main - 2026
- Mathematics
- Inverse Trigonometric Functions
Questions Asked in JEE Main exam
- Identify the correct statements: The presence of –NO\(_2\) group in benzene ring A. activates the ring towards electrophilic substitutions. B. deactivates the ring towards electrophilic substitutions. C. activates the ring towards nucleophilic substitutions. D. deactivates the ring towards nucleophilic substitutions. Choose the correct answer from the options given below:
- JEE Main - 2026
- General Chemistry
- Let $\vec{a}=2\hat{i}-\hat{j}-\hat{k}$, $\vec{b}=\hat{i}+3\hat{j}-\hat{k}$ and $\vec{c}=2\hat{i}+\hat{j}+3\hat{k}$. Let $\vec{v}$ be the vector in the plane of $\vec{a}$ and $\vec{b}$, such that the length of its projection on the vector $\vec{c}$ is $\dfrac{1}{\sqrt{14}}$. Then $|\vec{v}|$ is equal to
- JEE Main - 2026
- Vector Algebra
A substance 'X' (1.5 g) dissolved in 150 g of a solvent 'Y' (molar mass = 300 g mol$^{-1}$) led to an elevation of the boiling point by 0.5 K. The relative lowering in the vapour pressure of the solvent 'Y' is $____________ \(\times 10^{-2}\). (nearest integer)
[Given : $K_{b}$ of the solvent = 5.0 K kg mol$^{-1}$]
Assume the solution to be dilute and no association or dissociation of X takes place in solution.- JEE Main - 2026
- Solutions
- The wavelength of photon 'A' is 400 nm. The frequency of photon 'B' is \(10^{16}\,\text{s}^{-1}\). The wave number of photon 'C' is \(10^{5}\,\text{cm}^{-1}\). The correct order of energy of these photons is:
- JEE Main - 2026
- General Chemistry
- Given below are two statements: Statement I: The increasing order of boiling point of hydrogen halides is HCl<HBr<HI<HF. Statement II: The increasing order of melting point of hydrogen halides is HCl<HBr<HF<HI. In the light of the above statements, choose the correct answer from the options given below:
- JEE Main - 2026
- General Chemistry
Concepts Used:
Inverse Trigonometric Functions
The inverse trigonometric functions are also called arcus functions or anti trigonometric functions. These are the inverse functions of the trigonometric functions with suitably restricted domains. Specifically, they are the inverse functions of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle’s trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.
Domain and Range Of Inverse Functions
Considering the domain and range of the inverse functions, following formulas are important to be noted:
- sin(sin−1x) = x, if -1 ≤ x ≤ 1
- cos(cos−1x) = x, if -1 ≤ x ≤ 1
- tan(tan−1x) = x, if -∞ ≤ x ≤∞
- cot(cot−1x) = x, if -∞≤ x ≤∞
- sec(sec−1x) = x, if -∞ ≤ x ≤ -1 or 1 ≤ x ≤ ∞
- cosec(cosec−1x) = x, if -∞ ≤ x ≤ -1 or 1 ≤ x ≤ ∞
Also, the following formulas are defined for inverse trigonometric functions.
- sin−1(sin y) = y, if -π/2 ≤ y ≤ π/2
- cos−1(cos y) =y, if 0 ≤ y ≤ π
- tan−1(tan y) = y, if -π/2 <y< π/2
- cot−1(cot y) = y if 0<y< π
- sec−1(sec y) = y, if 0 ≤ y ≤ π, y ≠ π/2
cosec−1(cosec y) = y if -π/2 ≤ y ≤ π/2, y ≠ 0