Inverse Trigonometric Function
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
3.0 Properties
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
Property 1
When angle is (-ve) i.e $(-x)$
(i) ${\sin ^{ - 1}}( - x) = - {\sin ^{ - 1}}x$ where $x \in [ - 1,1]$
(ii) $ {\cos ^{ - 1}}( - x) = \pi - {\cos ^{ - 1}}x$ where $x \in [ - 1,1]$
(iii) $ {\tan ^{ - 1}}( - x) = - {\tan ^{ - 1}}x,x \in R$
(iv) $ \cos e{c^{ - 1}}( - x) = - \cos e{c^{ - 1}}x$ where $\left| x \right| \geqslant 1$
(v) $ se{c^{ - 1}}( - x) = \pi - se{c^{ - 1}}x$ where $\left| x \right| \geqslant 1$
(vi) ${\cot ^{ - 1}}( - x) = \pi - {\cot ^{ - 1}}x$ where $x \in R$
Property 2
(i) $${\sin ^{ - 1}}\left( {\frac{1}{x}} \right) = {\text{cose}}{{\text{c}}^{ - 1}}x$$ where $x \geqslant 1,x \leqslant - 1$
(ii) $${\cos ^{ - 1}}\left( {\frac{1}{x}} \right) = {\sec ^{ - 1}}x$$ where $x \geqslant 1,x \leqslant - 1$
(iii) \[{\tan ^{ - 1}}\left( {\frac{1}{x}} \right) = \left\{ \begin{gathered} {\cot ^{ - 1}}x;x > 0 \hspace{1em} \\ - \pi + {\cot ^{ - 1}}x;x < 0 \hspace{1em} \\ \end{gathered} \right\}\]
Property 3
(i) $\sin \left( {{{\sin }^{ - 1}}x} \right) = x$ where $ - 1 \leqslant x \leqslant 1$
(ii) $\cos \left( {{{\cos }^{ - 1}}x} \right) = x; - 1 \leqslant x \leqslant 1$
(iii) $\tan \left( {{{\tan }^{ - 1}}x} \right) = x;x \in R$
(iv) ${\text{cosec}}\left( {{\text{cose}}{{\text{c}}^{ - 1}}x} \right) = x;\left| x \right| > 1$
(v) $\sec \left( {{{\sec }^{ - 1}}x} \right) = x;x \leqslant - 1,x \geqslant 1$
(vi) $\cot \left( {{{\cot }^{ - 1}}x} \right) = x;x \in R$
(i) ${\sin ^{ - 1}}(\sin x) = $
\[\left\{ {\begin{array}{c} x&{0 \leqslant x \leqslant \frac{\pi }{2}} \\ {\pi - x}&{\frac{\pi }{2} \leqslant x \leqslant \pi } \\ {\pi - x}&{\pi \leqslant x \leqslant \frac{{3\pi }}{2}} \\ { - 2\pi + x}&{\frac{{3\pi }}{2} \leqslant x \leqslant 2\pi } \end{array}} \right.\]
(ii) ${\cos ^{ - 1}}(\cos \,\,x) = $
\[\left\{ {\begin{array}{c} x&{0 \leqslant x \leqslant \frac{\pi }{2}} \\ x&{\frac{\pi }{2} \leqslant x \leqslant \pi } \\ {2\pi - x}&{\pi \leqslant x \leqslant \frac{{3\pi }}{2}} \\ {2\pi - x}&{\frac{{3\pi }}{2} \leqslant x \leqslant 2\pi } \end{array}} \right.\]
(iii) ${\tan ^{ - 1}}(\tan x)=$
\[\left\{ {\begin{array}{c} x&{0 \leqslant x \leqslant \frac{\pi }{2}} \\ {x - \pi }&{\frac{\pi }{2} \leqslant x \leqslant \pi } \\ {x - \pi }&{\pi \leqslant x \leqslant \frac{{3\pi }}{2}} \\ {x - 2\pi }&{\frac{{3\pi }}{2} \leqslant x \leqslant 2\pi } \end{array}} \right.\]
(iv) ${\text{cose}}{{\text{c}}^{ - 1}}({\text{cosec}}\;x)$. The graph of this function is similar to ${\sin ^{ - 1}}(\sin x)$
(v) $se{c^{ - 1}}(secx)$. The graph of this function is similar to ${\cos ^{ - 1}}(\cos x)$
(vi) ${\cot ^{ - 1}}(\cot x)$. The graph of this function is not similar to ${\tan ^{ - 1}}(\tan x)$. It consists only the positive part of ${\tan ^{ - 1}}(\tan x)$
Property 5
(i) ${\sin ^{ - 1}}x + {\cos ^{ - 1}}x = \frac{\pi }{2}; - 1 \leqslant x \leqslant 1$
(ii) ${\tan ^{ - 1}}x + {\cot ^{ - 1}}x = \frac{\pi }{2};x \in R$
(iii) ${\text{cose}}{{\text{c}}^{ - 1}}x + {\sec ^{ - 1}}x = \frac{\pi }{2};\;\left| x \right| \geqslant 1$
Property 6
(i) ${\tan ^{ - 1}}x + {\tan ^{ - 1}}y$\[= \left\{ \begin{gathered} {\tan ^{ - 1}}\frac{{x + y}}{{1 - xy}};xy < 1 \hspace{1em} \\ \pi + {\tan ^{ - 1}}\frac{{x + y}}{{1 - xy}};x > 0,y > 0,xy > 1 \hspace{1em} \\ - \pi + {\tan ^{ - 1}}\frac{{x + y}}{{1 - xy}};x < 0,y < 0,xy > 1 \hspace{1em} \\ \end{gathered} \right\}\]
(ii) ${\tan ^{ - 1}}x - {\tan ^{ - 1}}y$\[= \left\{ \begin{gathered} {\tan ^{ - 1}}\left( {\frac{{x - y}}{{1 + xy}}} \right);xy > - 1 \hspace{1em} \\ \pi + {\tan ^{ - 1}}\left( {\frac{{x - y}}{{1 + xy}}} \right);x > 0,y < 0,xy < - 1 \hspace{1em} \\ - \pi + {\tan ^{ - 1}}\left( {\frac{{x - y}}{{1 + xy}}} \right);x < 0,y > 0,xy < - 1 \hspace{1em} \\ \end{gathered} \right\}\]
Property 7
(i) ${\sin ^{ - 1}}x + {\sin ^{ - 1}}y$
\[ = \left\{ {\begin{array}{c} {{{\sin }^{ - 1}}\left( {x\sqrt {1 - {y^2}} + y\sqrt {1 - {x^2}} } \right)}&{x \geqslant 0,y \geqslant 0,{x^2} + {y^2} \leqslant 1} \\ {\pi - {{\sin }^{ - 1}}\left( {x\sqrt {1 - {y^2}} + y\sqrt {1 - {x^2}} } \right)}&{x \geqslant 0,y \geqslant 0,{x^2} + {y^2} \geqslant 1} \end{array}} \right.\]
(ii) ${\sin ^{ - 1}}x - {\sin ^{ - 1}}y$
\[ = \left\{ {\begin{array}{c} {{{\sin }^{ - 1}}\left( {x\sqrt {1 - {y^2}} - y\sqrt {1 - {x^2}} } \right)}&{x,y \in [0,1]} \end{array}} \right.\]
Property 8
(i) ${\cos ^{ - 1}}x + {\cos ^{ - 1}}y$
\[ = \left\{ {\begin{array}{c} {{{\cos }^{ - 1}}\left\{ {xy - \sqrt {1 - {x^2}} \sqrt {1 - {y^2}} } \right\}}&{x,y \in [0,1]} \end{array}} \right.\]
(ii) ${\cos ^{ - 1}}x - {\cos ^{ - 1}}y$
\[ = \left\{ {\begin{array}{c} {{{\cos }^{ - 1}}\left\{ {xy + \sqrt {1 - {x^2}} \sqrt {1 - {y^2}} } \right\}}&{0 \leqslant x < y \leqslant 1} \\ { - {{\cos }^{ - 1}}\{ xy + \sqrt {1 - {x^2}} \sqrt {1 - {y^2}} }&{0 \leqslant y < x \leqslant 1} \end{array}} \right.\]
Property 9
$$\begin{equation} \begin{aligned} (i){\sin ^{ - 1}}x = {\cos ^{ - 1}}\left( {\sqrt {1 - {x^2}} } \right) = {\tan ^{ - 1}}\left( {\frac{x}{{\sqrt {1 - {x^2}} }}} \right) = {\cot ^{ - 1}}\left( {\frac{{\sqrt {1 - {x^2}} }}{x}} \right) = {\sec ^{ - 1}}\left( {\frac{1}{{\sqrt {1 - {x^2}} }}} \right) = \cos e{c^{ - 1}}\left( {\frac{1}{x}} \right) \\ (ii){\cos ^{ - 1}}x = {\sin ^{ - 1}}\left( {\sqrt {1 - {x^2}} } \right) = {\tan ^{ - 1}}\left( {\frac{{\sqrt {1 - {x^2}} }}{x}} \right) = {\cot ^{ - 1}}\left( {\frac{x}{{\sqrt {1 - {x^2}} }}} \right) = se{c^{ - 1}}\left( {\frac{1}{x}} \right) = \cos e{c^{ - 1}}\left( {\frac{1}{{\sqrt {1 - {x^2}} }}} \right) \\ (iii){\tan ^{ - 1}}x = {\sin ^{ - 1}}\left( {\frac{x}{{\sqrt {1 + {x^2}} }}} \right) = {\cos ^{ - 1}}\left( {\frac{1}{{\sqrt {1 + {x^2}} }}} \right) = {\cot ^{ - 1}}\left( {\frac{1}{x}} \right) = \cos e{c^{ - 1}}\left( {\frac{{\sqrt {1 + {x^2}} }}{x}} \right) = {\sec ^{ - 1}}\left( {\sqrt {1 + {x^2}} } \right) \\\end{aligned} \end{equation} $$
Property 10
\[\begin{gathered} (i)2{\sin ^{ - 1}}x = \left\{ \begin{gathered} {\sin ^{ - 1}}(2x\sqrt {1 - {x^2}} ;\frac{{ - 1}}{{\sqrt 2 }} \leqslant x \leqslant \frac{1}{{\sqrt 2 }} \hspace{1em} \\ \pi - {\sin ^{ - 1}}(2x\sqrt {1 - {x^2}} ;\frac{1}{{\sqrt 2 }} \leqslant x \leqslant 1 \hspace{1em} \\ - \pi - {\sin ^{ - 1}}(2x\sqrt {1 - {x^2}} ; - 1 \leqslant x \leqslant - \frac{1}{{\sqrt 2 }} \hspace{1em} \\ \end{gathered} \right\} \hspace{1em} \\ (ii)2{\cos ^{ - 1}}x = \left\{ \begin{gathered} {\cos ^{ - 1}}(2{x^2} - 1);0 \leqslant x \leqslant 1 \hspace{1em} \\ 2\pi - {\cos ^{ - 1}}(2{x^2} - 1); - 1 \leqslant x \leqslant 0 \hspace{1em} \\ \end{gathered} \right\} \hspace{1em} \\ (iii)2{\tan ^{ - 1}}x = \left\{ \begin{gathered} {\tan ^{ - 1}}\left( {\frac{{2x}}{{1 - {x^2}}}} \right); - 1 < x \leqslant 1 \hspace{1em} \\ \pi + {\tan ^{ - 1}}\left( {\frac{{2x}}{{1 - {x^2}}}} \right);x > 1 \hspace{1em} \\ - \pi + {\tan ^{ - 1}}\left( {\frac{{2x}}{{1 - {x^2}}}} \right);x < - 1 \hspace{1em} \\ \end{gathered} \right\} \hspace{1em} \\ \end{gathered} \]
Property 11
\[\begin{gathered} (i)3{\sin ^{ - 1}}x = \left\{ \begin{gathered} {\sin ^{ - 1}}(3x - 4{x^3});\frac{{ - 1}}{2} \leqslant x \leqslant \frac{1}{2} \hspace{1em} \\\pi - {\sin ^{ - 1}}(3x - 4{x^3});\frac{1}{2} \leqslant x \leqslant 1 \hspace{1em} \\ - \pi - {\sin ^{ - 1}}(3x - 4{x^3}); - 1 \leqslant x \leqslant \frac{{ - 1}}{2} \hspace{1em} \\ \end{gathered} \right\} \hspace{1em} \\ (ii)3{\cos ^{ - 1}}x = \left\{ \begin{gathered} {\cos ^{ - 1}}(4{x^3} - 3x);\frac{1}{2} \leqslant x \leqslant 1 \hspace{1em} \\ 2\pi - {\cos ^{ - 1}}(4{x^3} - 3x);\frac{{ - 1}}{2} \leqslant x \leqslant \frac{1}{2} \hspace{1em} \\ 2\pi + {\cos ^{ - 1}}(4{x^3} - 3x); - 1 \leqslant x \leqslant \frac{{ - 1}}{2} \hspace{1em} \\ \end{gathered} \right\} \hspace{1em} \\ (iii)3{\tan ^{ - 1}}x = \left\{ \begin{gathered} {\tan ^{ - 1}}\left( {\frac{{3x - {x^3}}}{{1 - 3{x^2}}}} \right);\frac{{ - 1}}{{\sqrt 3 }} \leqslant x \leqslant \frac{1}{{\sqrt 3 }} \hspace{1em} \\ \pi + {\tan ^{ - 1}}\left( {\frac{{3x - {x^3}}}{{1 - 3{x^2}}}} \right);x \geqslant \frac{1}{{\sqrt 3 }} \hspace{1em} \\ - \pi + {\tan ^{ - 1}}\left( {\frac{{3x - {x^3}}}{{1 - 3{x^2}}}} \right);x \leqslant \frac{{ - 1}}{{\sqrt 3 }} \hspace{1em} \\ \end{gathered} \right\} \hspace{1em} \\ \end{gathered} \]
Property 12
\[\begin{gathered} (i)2{\tan ^{ - 1}}x = \left\{ \begin{gathered} {\sin ^{ - 1}}\left( {\frac{{2x}}{{1 + {x^2}}}} \right); - 1 \leqslant x \leqslant 1 \hspace{1em} \\ \pi - {\sin ^{ - 1}}\left( {\frac{{2x}}{{1 + {x^2}}}} \right);x > 1 \hspace{1em} \\ - \pi - {\sin ^{ - 1}}\left( {\frac{{2x}}{{1 + {x^2}}}} \right);x < - 1 \hspace{1em} \\\end{gathered} \right\} \hspace{1em} \\(ii)2{\tan ^{ - 1}}x = \left\{ \begin{gathered} {\cos ^{ - 1}}\left( {\frac{{1 - {x^2}}}{{1 + {x^2}}}} \right);0 \leqslant x \leqslant \infty \hspace{1em} \\ - {\cos ^{ - 1}}\left( {\frac{{1 - {x^2}}}{{1 + {x^2}}}} \right); - \infty \leqslant x \leqslant 0 \hspace{1em} \\ \end{gathered} \right\} \hspace{1em} \\ \end{gathered} \]
Property 13
$(i) {\tan ^{ - 1}}x + {\tan ^{ - 1}}y + {\tan ^{ - 1}}z = {\tan ^{ - 1}}\left( {\frac{{x + y + z - xyz}}{{1 - xy - yz - zx}}} \right)$
$(ii)$ If ${\tan ^{ - 1}}x + {\tan ^{ - 1}}y + {\tan ^{ - 1}}z = \frac{\pi }{2}$ then $xy + yz + zx = 1$
$(iii)$ If ${\tan ^{ - 1}}x + {\tan ^{ - 1}}y + {\tan ^{ - 1}}z = \pi $ then $x + y + z = xyz$
$(iv)$ If ${\sin ^{ - 1}}x + {\sin ^{ - 1}}y + {\sin ^{ - 1}}z = \frac{\pi }{2}$ then ${x^2} + {y^2} + {z^2} + 2xyz = 1$
$(v)$ If ${\sin ^{ - 1}}x + {\sin ^{ - 1}}y + {\sin ^{ - 1}}z = \pi $ then $x\sqrt {1 - {x^2}} + y\sqrt {1 - {y^2}} + z\sqrt {1 - {z^2}} = 2xyz$
$(vi)$ If ${\cos ^{ - 1}}x + {\cos ^{ - 1}}y + {\cos ^{ - 1}}z = 3\pi $, then $xy + yz + zx = 3$
$(vii)$ If ${\cos ^{ - 1}}x + {\cos ^{ - 1}}y + {\cos ^{ - 1}}z = \pi $ then ${x^2} + {y^2} + {z^2} + 2xyz = 1$
$(vii)$ If ${\sin ^{ - 1}}x + {\sin ^{ - 1}}y + {\sin ^{ - 1}}z = \frac{{3\pi }}{2}$ then $xy + yz + zx = 3$
$(viii)$ If ${\sin ^{ - 1}}x + {\sin ^{ - 1}}y = \theta $ then ${\cos ^{ - 1}}x + {\cos ^{ - 1}}y = \pi - \theta $
$(ix)$ If ${\cos ^{ - 1}}x + {\cos ^{ - 1}}y = \theta $ then ${\sin ^{ - 1}}x + {\sin ^{ - 1}}y = \pi - \theta $
$(x)$ If ${\tan ^{ - 1}}x + {\tan ^{ - 1}}y = \frac{\pi }{2}$ then $xy=1$
$(xi)$ If ${\cot ^{ - 1}}x + {\cot ^{ - 1}}y = \frac{\pi }{2}$ then $xy=1$
$(xii)$ If ${\cos ^{ - 1}}\frac{x}{a} + {\cos ^{ - 1}}\frac{y}{b} = \theta $ then ${\left( {\frac{x}{a}} \right)^2} - \frac{{2xy}}{{ab}}\cos \theta + {\left( {\frac{y}{b}} \right)^2} = {\sin ^2}\theta $
$(xiii)$ ${\tan ^{ - 1}}{x_1} + {\tan ^{ - 1}}{x_2} + {\tan ^{ - 1}}{x_3}... + {\tan ^{ - 1}}{x_n} = {\tan ^{ - 1}}\left( {\frac{{{S_1} - {S_3} + {S_{5 - ..}}}}{{1 - {S_2} + {S_4} - {S_6} + ...}}} \right)$
where ${{S_k}}$ denotes the sum of products of ${{x_1}}$, ${{x_2}}$, ${{x_3}}$ taken $k$ at a time.
