Maths > Inverse Trigonometric Function > 3.0 Properties
Inverse Trigonometric Function
1.0 Introduction
2.0 Inverse Trigonometric function
2.1 ${\sin ^{ - 1}}x$:
2.2 ${\cos ^{ - 1}}x$:
2.3 ${\tan ^{ - 1}}x$:
2.4 ${\text{cose}}{{\text{c}}^{ - 1}}x$:
2.5 ${\sec^{ - 1}}x$
2.6 ${\cot^{ - 1}}x$
2.7 Summary
3.0 Properties
3.1 Property 1
3.2 Property 2
3.3 Property 3
3.4 Property 4
3.5 Property 5
3.6 Property 6
3.7 Property 7
3.8 Property 8
3.9 Property 9
3.10 Property 10
3.11 Property 11
3.12 Property 12
3.13 Property 13
3.10 Property 10
\[\begin{gathered} (i)2{\sin ^{ - 1}}x = \left\{ \begin{gathered} {\sin ^{ - 1}}(2x\sqrt {1 - {x^2}} ;\frac{{ - 1}}{{\sqrt 2 }} \leqslant x \leqslant \frac{1}{{\sqrt 2 }} \hspace{1em} \\ \pi - {\sin ^{ - 1}}(2x\sqrt {1 - {x^2}} ;\frac{1}{{\sqrt 2 }} \leqslant x \leqslant 1 \hspace{1em} \\ - \pi - {\sin ^{ - 1}}(2x\sqrt {1 - {x^2}} ; - 1 \leqslant x \leqslant - \frac{1}{{\sqrt 2 }} \hspace{1em} \\ \end{gathered} \right\} \hspace{1em} \\ (ii)2{\cos ^{ - 1}}x = \left\{ \begin{gathered} {\cos ^{ - 1}}(2{x^2} - 1);0 \leqslant x \leqslant 1 \hspace{1em} \\ 2\pi - {\cos ^{ - 1}}(2{x^2} - 1); - 1 \leqslant x \leqslant 0 \hspace{1em} \\ \end{gathered} \right\} \hspace{1em} \\ (iii)2{\tan ^{ - 1}}x = \left\{ \begin{gathered} {\tan ^{ - 1}}\left( {\frac{{2x}}{{1 - {x^2}}}} \right); - 1 < x \leqslant 1 \hspace{1em} \\ \pi + {\tan ^{ - 1}}\left( {\frac{{2x}}{{1 - {x^2}}}} \right);x > 1 \hspace{1em} \\ - \pi + {\tan ^{ - 1}}\left( {\frac{{2x}}{{1 - {x^2}}}} \right);x < - 1 \hspace{1em} \\ \end{gathered} \right\} \hspace{1em} \\ \end{gathered} \]2.2 ${\cos ^{ - 1}}x$:
2.3 ${\tan ^{ - 1}}x$:
2.4 ${\text{cose}}{{\text{c}}^{ - 1}}x$:
2.5 ${\sec^{ - 1}}x$
2.6 ${\cot^{ - 1}}x$
2.7 Summary
3.2 Property 2
3.3 Property 3
3.4 Property 4
3.5 Property 5
3.6 Property 6
3.7 Property 7
3.8 Property 8
3.9 Property 9
3.10 Property 10
3.11 Property 11
3.12 Property 12
3.13 Property 13
