JEE Main · 2024 · 09 Apr 2024 · Shift 2 · Inverse Trigonometric Functions

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation

2 sin^{- 1} x + 3 cos^{- 1} x = \frac{2 π}{5}

, is ____.

Answer: 0

Solution

2 sin^{- 1} x + 3 cos^{- 1} x = \frac{2 π}{5}

. Write it as

2 (sin^{- 1} x + cos^{- 1} x) + cos^{- 1} x = \frac{2 π}{5}

, and use

sin^{- 1} x + cos^{- 1} x = \frac{π}{2}

π + cos^{- 1} x = \frac{2 π}{5} ⟹ cos^{- 1} x = \frac{2 π}{5} - π = - \frac{3 π}{5}

Since the principal range of

cos^{- 1}

[0, π]

, a negative value

- \frac{3 π}{5}

is never attained.

Number of real solutions

= 0

JEE Main · 2024 · 09 Apr 2024 · Shift 2 · Integer answer

If the domain of the function $f (x) = cos^{- 1} (\frac{2 x - 5}{11 - 3 x}) + sin^{- 1} (2 x^{2} - 3 x + 1)$ is the interval $[α, β]$ , then $α + 2 β$ is equal to :
JEE Main 2026View solution →
The number of solutions of $tan^{- 1} 4 x + tan^{- 1} 6 x = \frac{π}{6}$ , where $- \frac{1}{2 6} < x < \frac{1}{2 6}$ is equal to
JEE Main 2026View solution →
If the domain of the function $f (x) = sin^{- 1} (\frac{1}{x ^{2} - 2 x - 2})$ , is $(- \infty, α] \cup [β, γ] \cup [δ, \infty)$ , then $α + β + γ + δ$ is equal to
JEE Main 2026View solution →
If $k = tan (\frac{π}{4} + \frac{1}{2} cos^{- 1} (\frac{2}{3})) + tan (\frac{1}{2} sin^{- 1} (\frac{2}{3}))$ then the number of solutions of the equation …
JEE Main 2026View solution →
Considering the principal values of inverse trigonometric functions, the value of the expression $tan (2 sin^{- 1} (\frac{2}{13}) - 2 cos^{- 1} (\frac{3}{10}))$ is equal to :
JEE Main 2026View solution →
Let [ • ] denote the greatest integer function. If the domain of the function $f (x) = cos^{- 1} (\frac{4 x + 2 [ x ]}{3})$ is $[α, β]$ , then $12 (α + β)$ is equal to :
JEE Main 2026View solution →