Roboguru

Jika tgα=32​, dan tgβ=−31​, , maka nilai cos(α−β)+sin(α−β)=…

Pertanyaan

Jika begin mathsize 14px style tg space alpha equals 2 over 3 comma space end styleundefined dan begin mathsize 14px style tg space beta equals negative 1 third comma end style Error converting from MathML to accessible text., maka nilai begin mathsize 14px style cos space open parentheses alpha minus beta close parentheses plus sin space open parentheses alpha minus beta close parentheses equals horizontal ellipsis end style

Pembahasan:

Diketahui begin mathsize 14px style tg space alpha equals 2 over 3 comma space end styleundefined. Karena tangen adalah perbandingan antara sisi depan dan sisi samping suatu segitiga siku-siku, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row miring equals cell square root of depan squared plus samping squared end root end cell row blank equals cell square root of 2 squared plus 3 squared end root end cell row blank equals cell square root of 4 plus 9 end root end cell row blank equals cell square root of 13 end cell end table end style

Akibatnya, karena undefined, nilai sinus dan cosinusnya adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell table row cell sin space alpha equals negative depan over miring end cell dan cell cos space alpha equals negative samping over miring end cell row cell equals negative fraction numerator 2 over denominator square root of 13 end fraction end cell blank cell equals negative fraction numerator 3 over denominator square root of 13 end fraction end cell end table end cell end table end style

Selanjutnya, diketahui begin mathsize 14px style tg space beta equals negative 1 third comma end style Error converting from MathML to accessible text.. Maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row miring equals cell square root of depan squared plus samping squared end root end cell row blank equals cell square root of 1 squared plus 3 squared end root end cell row blank equals cell square root of 1 plus 9 end root end cell row blank equals cell square root of 10 end cell end table end style

Akibatnya, karena Error converting from MathML to accessible text., nilai sinus dan cosinusnya adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell table row cell sin space beta equals negative depan over miring end cell dan cell cos space beta equals samping over miring end cell row cell equals negative fraction numerator 1 over denominator square root of 10 end fraction end cell blank cell equals fraction numerator 3 over denominator square root of 10 end fraction end cell end table end cell end table end style

Perhatikan bahwa

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell cos space left parenthesis italic alpha minus sign italic beta right parenthesis end cell equals cell cos space italic alpha space cos space italic beta and sin space italic alpha space sin space italic beta end cell row blank equals cell open parentheses negative sign fraction numerator 3 over denominator square root of 13 end fraction close parentheses open parentheses fraction numerator 3 over denominator square root of 10 end fraction close parentheses plus open parentheses negative sign fraction numerator 2 over denominator square root of 13 end fraction close parentheses open parentheses negative sign fraction numerator 1 over denominator square root of 10 end fraction close parentheses end cell row blank equals cell negative sign fraction numerator 9 over denominator square root of 130 end fraction plus fraction numerator 2 over denominator square root of 130 end fraction end cell row blank equals cell negative sign fraction numerator 7 over denominator square root of 130 end fraction end cell end table end style

dan

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell sin space left parenthesis italic alpha minus sign italic beta right parenthesis end cell equals cell sin space italic alpha space cos space italic beta italic minus sign cos space italic alpha space sin space italic beta end cell row blank equals cell open parentheses negative sign fraction numerator 2 over denominator square root of 13 end fraction close parentheses open parentheses fraction numerator 3 over denominator square root of 10 end fraction close parentheses minus sign open parentheses negative sign fraction numerator 3 over denominator square root of 13 end fraction close parentheses open parentheses negative sign fraction numerator 1 over denominator square root of 10 end fraction close parentheses end cell row blank equals cell negative sign fraction numerator 6 over denominator square root of 130 end fraction minus sign fraction numerator 3 over denominator square root of 130 end fraction end cell row blank equals cell negative sign fraction numerator 9 over denominator square root of 130 end fraction end cell end table end style

Dengan demikian

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell cos space left parenthesis alpha minus beta right parenthesis plus sin space left parenthesis alpha minus beta right parenthesis end cell equals cell open parentheses negative fraction numerator 7 over denominator square root of 130 end fraction close parentheses plus open parentheses negative fraction numerator 9 over denominator square root of 130 end fraction close parentheses end cell row blank equals cell negative fraction numerator 16 over denominator square root of 130 end fraction times fraction numerator square root of 130 over denominator square root of 130 end fraction end cell row blank equals cell negative fraction numerator 16 square root of 130 over denominator 130 end fraction end cell row blank equals cell negative 4 over 65 square root of 130 end cell end table end style

Jadi, begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell cos space left parenthesis alpha minus beta right parenthesis plus sin space left parenthesis alpha minus beta right parenthesis end cell end table equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 4 over 65 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 130 end cell end table end style.

Jawaban terverifikasi

Dijawab oleh:

W. Lestari

Mahasiswa/Alumni Universitas Sriwijaya

Terakhir diupdate 07 Oktober 2021

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