Maths · Inverse trigonometrical functions and their properties
The number of real solutions of the equation }+ \) is
The number of real solutions of the equation \( \tan ^{-1} \sqrt{x(x+1)}+ \) \( \sin ^{-1} \sqrt{x^{2}+x+1}=\frac{\pi}{2} \) is
- A. One
- B. Four
- C. Two
- D. Infinitely many
Step-by-step solution
The domain of the equation requires both sqrt(x(x+1)) (for arctan) and sqrt(x^2+x+1) (for arcsin) to be defined and within appropriate ranges. From arctan, x(x+1) ≥ 0 ⇒ x ≤ -1 or x ≥ 0. From arcsin, 0 ≤ √(x^2+x+1) ≤ 1 ⇒ x^2+x+1 ≤ 1 ⇒ x(x+1) ≤ 0 ⇒ -1 ≤ x ≤ 0. Intersection gives x = -1 or x = 0. Both satisfy the original equation: for x=0, arctan(0)+arcsin(1)=0+π/2=π/2; for x=-1, similarly. Hence exactly two real solutions.
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