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Sia ? 6 years, 2 months ago
Types of relations are :
- Empty Relation
- Universal Relation
- Inverse Relation
- Reflexive Relation
- Symmetric Relation
- Transitive Relation
- Equivalence Relation
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Posted by Dolly ?️ 6 years, 2 months ago
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Sia ? 6 years, 2 months ago
Given, {tex}{\cos ^{ - 1}}\frac{x}{a} + {\cos ^{ - 1}}\frac{y}{b} = a{/tex}
{tex}\left[ {\because {{\cos }^{ - 1}}x + {{\cos }^{ - 1}}y} \right. \left. { = {{\cos }^{ - 1}}\left( {xy - \sqrt {1 - {x^2}} \sqrt {1 - {y^2}} } \right)} \right]{/tex}
{tex}\Rightarrow {\cos ^{ - 1}}\left[ {\frac{x}{a}.\frac{y}{b} - \sqrt {1 - \frac{{{x^2}}}{{{a^2}}}} .\sqrt {1 - \frac{{{y^2}}}{{{b^2}}}} } \right] = a{/tex}
{tex}\Rightarrow\frac{{xy}}{{ab}} - \sqrt {1 - \frac{{{x^2}}}{{{a^2}}}} \cdot \sqrt {1 - \frac{{{y^2}}}{{{b^2}}}} = \cos a{/tex}
{tex}\Rightarrow \frac{{xy}}{{ab}} - \cos a = \sqrt {1 - \frac{{{x^2}}}{{{a^2}}}} \sqrt {1 - \frac{{{y^2}}}{{{b^2}}}}{/tex}
On squaring both side, we get
{tex}\Rightarrow{\left( {\frac{{xy}}{{ab}} - \cos a } \right)^2} = {\left( {\sqrt {1 - \frac{{{x^2}}}{{{a^2}}}} \sqrt {1 - \frac{{{y^2}}}{{{b^2}}}} } \right)^2}{/tex}
{tex}\Rightarrow \frac{{{x^2}{y^2}}}{{{a^2}{b^2}}} + {\cos ^2}a - 2.\frac{{xy}}{{ab}}.\cos a = \left( {1 - \frac{{{x^2}}}{{{a^2}}}} \right)\left( {1 - \frac{{{y^2}}}{{{b^2}}}} \right){/tex}
{tex}\Rightarrow\frac{{{x^2}{y^2}}}{{{a^2}{b^2}}} + {\cos ^2}a - 2\frac{{xy}}{{ab}}\cos a = 1 - \frac{{{y^2}}}{{{b^2}}} - \frac{{{x^2}}}{{{a^2}}} + \frac{{{x^2}{y^2}}}{{{a^2}{b^2}}}{/tex}
{tex}\Rightarrow\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} - 2\frac{{xy}}{{ab}}\cos a = 1 - {\cos ^2}a{/tex}
{tex}\Rightarrow\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} - 2\frac{{xy}}{{ab}}\cos a = {\sin ^2}a{/tex}
Miss Mor 6 years, 2 months ago
Posted by Dolly ?️ 6 years, 2 months ago
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Sia ? 6 years, 2 months ago
Let {tex}{\tan ^{ - 1}}\frac{y}{2} = \theta{/tex}, where {tex}\theta \in \left( { - \frac{\pi }{2},\;\frac{\pi }{2}} \right){/tex}. So, {tex}\tan \theta = \frac{y}{2}{/tex}, which gives {tex}\sec \theta = \frac{{\sqrt {4 + {y^2}} }}{2}{/tex}.
Therefore, {tex}\sec \left( {{{\tan }^{ - 1}}\frac{y}{2}} \right) = \sec \theta = \frac{{\sqrt {4 + {y^2}} }}{2}{/tex}.
Posted by Dolly ?️ 6 years, 2 months ago
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Sia ? 6 years, 2 months ago
We have, {tex}4{\tan ^{ - 1}}\frac{1}{5} - {\tan ^{ - 1}}\frac{1}{{239}}{/tex}
{tex}= 2.2{\tan ^{ - 1}}\frac{1}{5} - {\tan ^{ - 1}}\frac{1}{{239}}{/tex}
{tex}= 2\left[ {{{\tan }^{ - 1}}\frac{{\frac{2}{5}}}{{1 - {{\left( {\frac{1}{5}} \right)}^2}}}} \right] - {\tan ^{ - 1}}\frac{1}{{239}}{/tex} {tex}\left[ {\because 2{{\tan }^{ - 1}}x = {{\tan }^{ - 1}}\left( {\frac{{2x}}{{1 - {x^2}}}} \right)} \right]{/tex}
{tex} = 2.\left[ {{{\tan }^{ - 1}}\left( {\frac{{\frac{2}{5}}}{{1 - \frac{1}{{25}}}}} \right)} \right] - {\tan ^{ - 1}}\frac{1}{{239}}{/tex}
{tex} = 2.\left[ {{{\tan }^{ - 1}}\left( {\frac{{\frac{2}{5}}}{{\frac{{24}}{{25}}}}} \right)} \right] - {\tan ^{ - 1}}\frac{1}{{239}}{/tex}
{tex}= 2{\tan ^{ - 1}}\frac{5}{{12}} - ta{n^{ - 1}}\frac{1}{{239}}{/tex}
{tex}= {\tan ^{ - 1}}\frac{{2.\frac{5}{{12}}}}{{1 - {{\left( {\frac{5}{{12}}} \right)}^2}}} - {\tan ^{ - 1}}\frac{1}{{239}}{/tex} {tex}\left[ {\because 2{{\tan }^{ - 1}}x = {{\tan }^{ - 1}}\left( {\frac{{2x}}{{1 - {x^2}}}} \right)} \right]{/tex}
{tex} = {\tan ^{ - 1}}\left( {\frac{{\frac{5}{6}}}{{1 - \frac{{25}}{{144}}}}} \right) - {\tan ^{ - 1}}\frac{1}{{239}}{/tex}
{tex} = {\tan ^{ - 1}}\left( {\frac{{144 \times 5}}{{119 \times 6}}} \right) - {\tan ^{ - 1}}\frac{1}{{239}}{/tex}
{tex} = {\tan ^{ - 1}}\left( {\frac{{120}}{{199}}} \right) - {\tan ^{ - 1}}\frac{1}{{239}}{/tex}
{tex} = {\tan ^{ - 1}}\left( {\frac{{\frac{{120}}{{119}} - \frac{1}{{239}}}}{{1 + \frac{{120}}{{119}} \cdot \frac{1}{{239}}}}} \right){/tex}{tex}\left[ {\because \;{{\tan }^{ - 1}}x - {{\tan }^{ - 1}}y = {{\tan }^{ - 1}}\left( {\frac{{x - y}}{{1 + xy}}} \right)} \right]{/tex}
{tex} = {\tan ^{ - 1}}\left( {\frac{{120 \times 239 - 119}}{{119 \times 239 \times 120}}} \right){/tex}
{tex} = {\tan ^{ - 1}}\left[ {\frac{{28680 - 119}}{{28441 + 120}}} \right] = {\tan ^{ - 1}}\frac{{28561}}{{28561}}{/tex}
{tex}= {\tan ^{ - 1}}(1) = {\tan ^{ - 1}}\left( {\tan \frac{\pi }{4}} \right) = \frac{\pi }{4}{/tex}
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Silent Man 6 years, 2 months ago
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