Find the domain of the inverse trigonometric function \({\sin ^{ – 1}}\left( {2x\sqrt {1 – {x^2}} } \right)\) is,
1. \(\left[ { – 1,1} \right]\)
2. \(\left[ {0,\frac{1}{2}} \right]\)
3. \(\left[ { – \frac{1}{{\sqrt 2 }},\frac{1}{{\sqrt 2 }}} \right]\)
4. \(\left[ { – \frac{1}{2},\frac{1}{2}} \right]\)
1. \(\left[ { – 1,1} \right]\)
2. \(\left[ {0,\frac{1}{2}} \right]\)
3. \(\left[ { – \frac{1}{{\sqrt 2 }},\frac{1}{{\sqrt 2 }}} \right]\)
4. \(\left[ { – \frac{1}{2},\frac{1}{2}} \right]\)
Correct Answer – Option 3 : \(\left[ { – \frac{1}{{\sqrt 2 }},\frac{1}{{\sqrt 2 }}} \right]\)
Concept:
The domain of inverse sine function, sin x is \(x \in \left[ { – 1,1} \right]\)
Calculation:
Domain of the function is calculated as follows:
\({\sin ^{ – 1}}\left( {2x\sqrt {1 – {x^2}} } \right)\)
\(- 1 \le 2x\sqrt {1 – {x^2}} \le 1\)
\( – \frac{1}{2} \le x\sqrt {1 – {x^2}} \le \frac{1}{2}\)
\({x^2}\left( {1 – {x^2}} \right) \le \frac{1}{4}\)
\(t – {t^2} – \frac{1}{4} \le 0\)
\({\left( {t – \frac{1}{2}} \right)^2} \le 0\)
\(t \le \frac{1}{2}\)
\({x^2} \le \frac{1}{{\sqrt 2 }}\)
\(x \in \left[ { – \frac{1}{{\sqrt 2 }},\frac{1}{{\sqrt 2 }}} \right]\)