The minimum value of x which satisfies the inequality (s i n- 1 x)2≥ (c o s- 1 x)2 is

Question

The minimum value of x which satisfies the inequality (s i n- 1 x)2≥ (c o s- 1 x)2 is (A) (1/√2) (B) (1/2) (C) (√3/2) (D) (1/√3)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="Solution" src="https://cdn.tardigrade.in/q/nta/m-inijoomq0plrcczc.jpg" /> (s i n- 1 x)2-(c o s- 1 x)2≥ 0 (s i n- 1 x + c o s- 1 x)(s i n- 1 x - c o s- 1 x)≥ 0 ⇒ sin- 1x≥ cos- 1x Clearly from graph x∈ [(1/√2) , 1]

Q. The minimum value of $x$ which satisfies the inequality $(s i n^{- 1} x)^{2} \geq (co s^{- 1} x)^{2}$ is

$\frac{1}{2}$

$\frac{1}{2}$

$\frac{3}{2}$

$\frac{1}{3}$

$(s i n^{- 1} x)^{2} - (co s^{- 1} x)^{2} \geq 0$
$(s i n^{- 1} x + co s^{- 1} x) (s i n^{- 1} x - co s^{- 1} x) \geq 0$
$\Rightarrow s i n^{- 1} x \geq co s^{- 1} x$
Clearly from graph
$x \in [\frac{1}{2}, 1]$

Q. The minimum value of x which satisfies the inequality (sin−1x)2≥(cos−1x)2 is

2​1​

21​

23​​

3​1​

(sin−1x)2−(cos−1x)2≥0 (sin−1x+cos−1x)(sin−1x−cos−1x)≥0 ⇒sin−1x≥cos−1x Clearly from graph x∈[2​1​,1]

Q. The minimum value of $x$ which satisfies the inequality $(s i n^{- 1} x)^{2} \geq (co s^{- 1} x)^{2}$ is

$\frac{1}{2}$

$\frac{1}{2}$

$\frac{3}{2}$

$\frac{1}{3}$

$(s i n^{- 1} x)^{2} - (co s^{- 1} x)^{2} \geq 0$
$(s i n^{- 1} x + co s^{- 1} x) (s i n^{- 1} x - co s^{- 1} x) \geq 0$
$\Rightarrow s i n^{- 1} x \geq co s^{- 1} x$
Clearly from graph
$x \in [\frac{1}{2}, 1]$