If the inverse trigonometric functions take principal values, then
\(cos^{-1} ( \frac{3}{10} cos (tan^{-1} (\frac{4}{3})) + \frac{2}{5} sin (tan^{-1} (\frac{4}{3})))\)
is equal to :
If the inverse trigonometric functions take principal values, then
\(cos^{-1} ( \frac{3}{10} cos (tan^{-1} (\frac{4}{3})) + \frac{2}{5} sin (tan^{-1} (\frac{4}{3})))\)
is equal to :
0
\(\frac{\pi}{4}\)
\(\frac{\pi}{3}\)
\(\frac{\pi}{6}\)
The Correct Option is C
Solution and Explanation
The correct answer is (C) : \(\frac{\pi}{3}\)
\(cos^{-1} ( \frac{3}{10} cos (tan^{-1} (\frac{4}{3})) + \frac{2}{5} sin (tan^{-1} (\frac{4}{3})))\)
\(= cos^{-1} ( \frac{3}{10} . \frac{3}{5} + \frac{2}{5} . \frac{4}{5} )\)
\(= cos^{-1} (\frac{1}{2}) = \frac{π}{3}\)
Top Questions on Inverse Trigonometric Functions
- The total number of real solutions of the equation $$ \theta = \tan^{-1}(2 \tan \theta) - \frac{1}{2} \sin^{-1} \left( \frac{6 \tan \theta}{9 + \tan^2 \theta} \right) $$ is
(Here, the inverse trigonometric functions $ \sin^{-1} x $ and $ \tan^{-1} x $ assume values in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ and $(-\frac{\pi}{2}, \frac{\pi}{2})$, respectively.)- JEE Advanced - 2025
- Mathematics
- Inverse Trigonometric Functions
- Using the principal values of the inverse trigonometric functions the sum of the maximum and the minimum values of \(16((sec^{-1}x)^{2}+(cosec^{-1}x)^{2})\) is:
- JEE Main - 2025
- Mathematics
- Inverse Trigonometric Functions
- The number of solutions of \( \sin^{-1} x + \sin^{-1} (1 - x) = \cos^{-1} x \) is:
- WBJEE - 2025
- Mathematics
- Inverse Trigonometric Functions
- If \( \cos^{-1} \alpha + \cos^{-1} \beta + \cos^{-1} \gamma = 3\pi \), then the value of \( \alpha(\beta+\gamma) + \beta(\gamma+\alpha) + \gamma(\alpha+\beta) \) is:
- WBJEE - 2025
- Mathematics
- Inverse Trigonometric Functions
- If $ \tan^{-1}(x) = \tan^{-1}\left( 3 - \frac{\pi}{4} \right) $, find $ x $.
- KEAM - 2025
- Mathematics
- Inverse Trigonometric Functions
Questions Asked in JEE Main exam
- If the components of \( \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) along and perpendicular to \( \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) respectively are \( \frac{16}{11} (3\hat{i} + \hat{j} - \hat{k}) \) and \( \frac{1}{11} (-4\hat{i} - 5\hat{j} - 17\hat{k}) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to:
- JEE Main - 2025
- Vectors
If the domain of the function \( f(x) = \frac{1}{\sqrt{3x + 10 - x^2}} + \frac{1}{\sqrt{x + |x|}} \) is \( (a, b) \), then \( (1 + a)^2 + b^2 \) is equal to:
- JEE Main - 2025
- Functions
- Let \( A \) be a \( 3 \times 3 \) matrix such that \( X^T AX = 0 \) for all nonzero \( 3 \times 1 \) matrices \( X = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \). \[ A = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \quad A = \begin{bmatrix} 1 \\ 4 \\ -5 \end{bmatrix}, \quad A = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \quad A = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix} \] If \( \det(\text{adj}(2(A + I))) = 2^{\alpha} 3^{\beta} 5^{\gamma} \), \( \alpha, \beta, \gamma \in \mathbb{N} \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is:
In the given circuit the sliding contact is pulled outwards such that the electric current in the circuit changes at the rate of 8 A/s. At an instant when R is 12 Ω, the value of the current in the circuit will be A.- JEE Main - 2025
- Electrical Circuits
- Let \( f: [0, 3] \to A \) be defined by \( f(x) = 2x^3 - 15x^2 + 36x + 7 \) and \( g: [0, \infty) \to B \) be defined by \( g(x) = \frac{x}{x^{2025} + 1}. \) If both functions are onto and \( S = \{ x \in \mathbb{Z} : x \in A { or } x \in B \} \), then \( n(S) \) is equal to:
- JEE Main - 2025
- Functions
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