Buktikan bahwa hasil rotasi titik oleh adalah =...

Buktikan bahwa hasil rotasi titik $begin mathsize 14px style straight A open parentheses x comma y close parentheses end style$ oleh $begin mathsize 14px style straight R open square brackets straight O open parentheses 0 comma 0 close parentheses comma space theta close square brackets end style$ adalah $begin mathsize 14px style open parentheses fraction numerator x apostrophe over denominator y apostrophe end fraction close parentheses end style$ = $begin mathsize 14px style left parenthesis table row cell cos theta end cell cell negative sin theta end cell row cell sin theta end cell cell cos theta end cell end table right parenthesis space left parenthesis x over y right parenthesis end style$ .

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Jawaban:

Pada segitiga $size 14px O size 14px x size 14px A$ berlaku perbandingan trigonometri :

$begin mathsize 14px style x over r equals cos alpha left right double arrow x equals r space cos alpha y over r equals sin alpha left right double arrow y equals r space sin alpha end style$

Pada segitiga $size 14px O size 14px x size 14px apostrophe size 14px A size 14px apostrophe$ berlaku perbandingan trigonometri :

$fraction numerator size 14px x size 14px apostrophe over denominator size 14px r end fraction size 14px equals size 14px cos size 14px left parenthesis size 14px alpha size 14px plus size 14px theta size 14px right parenthesis size 14px left right double arrow size 14px x size 14px apostrophe size 14px equals size 14px r size 14px space size 14px cos size 14px left parenthesis size 14px alpha size 14px plus size 14px theta size 14px right parenthesis fraction numerator size 14px y size 14px apostrophe over denominator size 14px r end fraction size 14px equals size 14px sin size 14px left parenthesis size 14px alpha size 14px plus size 14px theta size 14px right parenthesis size 14px left right double arrow size 14px y size 14px apostrophe size 14px equals size 14px r size 14px space size 14px sin size 14px left parenthesis size 14px alpha size 14px plus size 14px theta size 14px right parenthesis$

Berdasarkan gambar di atas maka koordinat titik $begin mathsize 14px style straight A apostrophe left parenthesis x apostrophe comma y apostrophe right parenthesis end style$ yang merupakan bayangan titik $Error converting from MathML to accessible text.$ jika dirotasikan terhadap titik $Error converting from MathML to accessible text.$ sejauh sudut $theta$ . Sehingga dapat dituliskan bahwa :

$size 14px x size 14px apostrophe size 14px equals size 14px r size 14px space size 14px cos size 14px left parenthesis size 14px alpha size 14px plus size 14px theta size 14px right parenthesis size 14px x size 14px apostrophe size 14px equals size 14px r size 14px space size 14px left parenthesis size 14px cos size 14px space size 14px alpha size 14px space size 14px cos size 14px space size 14px theta size 14px minus size 14px sin size 14px space size 14px alpha size 14px space size 14px sin size 14px space size 14px theta size 14px right parenthesis size 14px x size 14px apostrophe size 14px equals size 14px r size 14px space size 14px cos size 14px space size 14px alpha size 14px space size 14px cos size 14px space size 14px theta size 14px minus size 14px r size 14px space size 14px sin size 14px space size 14px alpha size 14px space size 14px sin size 14px space size 14px theta size 14px x size 14px apostrophe size 14px equals size 14px x size 14px space size 14px cos size 14px space size 14px theta size 14px minus size 14px y size 14px space size 14px sin size 14px space size 14px theta size 14px y size 14px apostrophe size 14px equals size 14px r size 14px space size 14px sin size 14px left parenthesis size 14px alpha size 14px plus size 14px theta size 14px right parenthesis size 14px y size 14px apostrophe size 14px equals size 14px r size 14px space size 14px left parenthesis size 14px sin size 14px space size 14px alpha size 14px space size 14px cos size 14px space size 14px theta size 14px plus size 14px cos size 14px space size 14px alpha size 14px space size 14px sin size 14px space size 14px theta size 14px right parenthesis size 14px y size 14px apostrophe size 14px equals size 14px r size 14px space size 14px sin size 14px space size 14px alpha size 14px space size 14px cos size 14px space size 14px theta size 14px plus size 14px r size 14px space size 14px cos size 14px space size 14px alpha size 14px space size 14px sin size 14px space size 14px theta size 14px y size 14px apostrophe size 14px equals size 14px y size 14px space size 14px cos size 14px space size 14px theta size 14px plus size 14px x size 14px space size 14px sin size 14px space size 14px theta size 14px y size 14px apostrophe size 14px equals size 14px x size 14px space size 14px sin size 14px space size 14px theta size 14px plus size 14px y size 14px space size 14px cos size 14px space size 14px theta$

Jika ditulis dalam bentuk matriks akan menjadi :

$begin mathsize 14px style straight A apostrophe equals open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses equals open parentheses table row cell cos theta end cell cell negative sin theta end cell row cell sin theta end cell cell cos theta end cell end table close parentheses open parentheses table row x row y end table close parentheses end style$

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