Find the linear equations that can be used to convert an (x, y) equation to a (x, v) equation using the given angle of rotation \displaystyle\theta. \displaystyle\theta={{\tan}^{{-{1}}}{\left({5}\text{/}{12}\right)}}

Question
Trigonometry
asked 2021-01-04
Find the linear equations that can be used to convert an (x, y) equation to a (x, v) equation using the given angle of rotation \(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le\theta\).
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le\theta={\left\lbrace{\left\lbrace{\tan}\right\rbrace}^{{{\left\lbrace-{\left\lbrace{1}\right\rbrace}\right\rbrace}}}{\left\lbrace{\left({\left\lbrace{5}\right\rbrace}\text{/}{\left\lbrace{12}\right\rbrace}\right)}\right\rbrace}\right\rbrace}\)

Answers (1)

2021-01-05
Use the axis rotation formulas:
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{x}\right\rbrace}={\left\lbrace{u}\right\rbrace}{\cos{{\left\lbrace\theta\right\rbrace}}}-{\left\lbrace{v}\right\rbrace}{\sin{{\left\lbrace\theta\right\rbrace}}}\)
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{y}\right\rbrace}={\left\lbrace{u}\right\rbrace}{\sin{{\left\lbrace\theta\right\rbrace}}}+{\left\lbrace{v}\right\rbrace}{\cos{{\left\lbrace\theta\right\rbrace}}}\)
From the given angle, we know that:
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\tan{{\left\lbrace\theta\right\rbrace}}}={\frac{{{5}}}{{{\left\lbrace{12}\right\rbrace}}}}\)
From this tangent ratio, opp \(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le={\left\lbrace{5}\right\rbrace}{\left\lbrace\quad\text{and}\quad\right\rbrace}{\left\lbrace{a}\right\rbrace}{\left\lbrace{d}\right\rbrace}{\left\lbrace{j}\right\rbrace}={\left\lbrace{12}\right\rbrace}\text{so}\ {\left\lbrace{h}\right\rbrace}{\left\lbrace{y}\right\rbrace}{\left\lbrace{p}\right\rbrace}=\sqrt{{{\left\lbrace{\left\lbrace{\left({\left\lbrace{o}\right\rbrace}{\left\lbrace{p}\right\rbrace}{\left\lbrace{p}\right\rbrace}\right)}\right\rbrace}^{{{2}}}+{\left\lbrace{\left({\left\lbrace{a}\right\rbrace}{\left\lbrace{d}\right\rbrace}{\left\lbrace{j}\right\rbrace}\right)}\right\rbrace}^{{{2}}}\right\rbrace}}}\)
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le=\sqrt{{{\left\lbrace{\left\lbrace{5}\right\rbrace}^{{{2}}}+{\left\lbrace{12}\right\rbrace}^{{{2}}}\right\rbrace}}}\)
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le=\sqrt{{{\left\lbrace{169}\right\rbrace}}}={\left\lbrace{13}\right\rbrace}\) so:
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\sin{{\left\lbrace\theta\right\rbrace}}}={\frac{{{\left\lbrace{\left\lbrace{o}\right\rbrace}{\left\lbrace{p}\right\rbrace}{\left\lbrace{p}\right\rbrace}\right\rbrace}}}{{{\left\lbrace{\left\lbrace{h}\right\rbrace}{\left\lbrace{y}\right\rbrace}{\left\lbrace{p}\right\rbrace}\right\rbrace}}}}={\frac{{{5}}}{{{\left\lbrace{13}\right\rbrace}}}}\)
and
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\cos{{\left\lbrace\theta\right\rbrace}}}={\frac{{{\left\lbrace{\left\lbrace{a}\right\rbrace}{\left\lbrace{d}\right\rbrace}{\left\lbrace{j}\right\rbrace}\right\rbrace}}}{{{\left\lbrace{\left\lbrace{h}\right\rbrace}{\left\lbrace{y}\right\rbrace}{\left\lbrace{p}\right\rbrace}\right\rbrace}}}}={\frac{{{12}}}{{{\left\lbrace{13}\right\rbrace}}}}\)
For x,
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{x}\right\rbrace}={\left\lbrace{12}\right\rbrace}{\frac{{{u}}}{{{\left\lbrace{13}\right\rbrace}}}}-{\left\lbrace{5}\right\rbrace}{\frac{{{v}}}{{{\left\lbrace{13}\right\rbrace}}}}\)
For y,
\(\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{y}\right\rbrace}={\left\lbrace{5}\right\rbrace}{\frac{{{u}}}{{{\left\lbrace{13}\right\rbrace}}}}+{\left\lbrace{12}\right\rbrace}{\frac{{{v}}}{{{\left\lbrace{13}\right\rbrace}}}}\)
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