Pertanyaan

Diketahui sin θ = 2 a , dengan θ berada di kuadran III. Tentukan nilai dari 1 − tan 2 θ 2 tan θ = …

Diketahui $sin θ = 2 a$ , dengan $θ$ berada di kuadran III. Tentukan nilai dari $\frac{2 tan θ}{1 - tan ^{2} θ} = \dots$

F. Ayudhita

Master Teacher

Jawaban terverifikasi

Pembahasan

Karena nilai sin di kuadran III negatif, maka nilai a adalah negatif, sehingga Karena nilai tan di kuadran III adalah positif, sedangkan nilai sin dan cos di kuadran III adalah negatif, dan dari yg diketahui bahwa nilai sin negatif karena nilai a yang negatif, maka nilai tan perlu dikalikan negatif agar bernilai positif, sehingga maka

$begin mathsize 14px style sin theta equals de over mi equals fraction numerator 2 a over denominator 1 end fraction end style$

Karena nilai sin di kuadran III negatif, maka nilai a adalah negatif, sehingga

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell tan theta end cell equals cell de over sa end cell row blank equals cell fraction numerator 2 a over denominator square root of 1 minus 4 a squared end root end fraction end cell row blank equals cell fraction numerator 2 a square root of 1 minus 4 a squared end root over denominator 1 minus 4 a squared end fraction end cell end table end style$

Karena nilai tan di kuadran III adalah positif, sedangkan nilai sin dan cos di kuadran III adalah negatif, dan dari yg diketahui bahwa nilai sin negatif karena nilai a yang negatif, maka nilai tan perlu dikalikan negatif agar bernilai positif, sehingga

$begin mathsize 14px style tan theta equals negative fraction numerator 2 a square root of 1 minus 4 a squared end root over denominator 1 minus 4 a squared end fraction end style$

maka

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 2 tan theta over denominator 1 minus tan squared theta end fraction end cell equals cell fraction numerator 2 times open parentheses negative fraction numerator 2 a square root of 1 minus 4 a squared end root over denominator 1 minus 4 a squared end fraction close parentheses over denominator 1 minus open parentheses negative fraction numerator 2 a square root of 1 minus 4 a squared end root over denominator 1 minus 4 a squared end fraction close parentheses squared end fraction end cell row blank equals cell fraction numerator negative fraction numerator 4 a square root of 1 minus 4 a squared end root over denominator 1 minus 4 a squared end fraction over denominator 1 minus open parentheses 2 a square root of 1 minus 4 a squared end root close parentheses squared over left parenthesis 1 minus 4 a squared right parenthesis squared end fraction end cell row blank equals cell fraction numerator negative fraction numerator 4 a square root of 1 minus 4 a squared end root over denominator 1 minus 4 a squared end fraction over denominator fraction numerator left parenthesis 1 minus 4 a squared right parenthesis squared minus 4 a squared left parenthesis 1 minus 4 a squared right parenthesis over denominator left parenthesis 1 minus 4 a squared right parenthesis squared end fraction end fraction end cell row blank equals cell open parentheses negative fraction numerator 4 a square root of 1 minus 4 a squared end root over denominator 1 minus 4 a squared end fraction close parentheses times fraction numerator left parenthesis 1 minus 4 a squared right parenthesis squared over denominator left parenthesis 1 minus 4 a squared right parenthesis times left parenthesis open parentheses 1 minus 4 a squared close parentheses minus 4 a squared right parenthesis end fraction end cell row blank equals cell negative fraction numerator left parenthesis 4 a square root of 1 minus 4 a squared end root right parenthesis times left parenthesis 1 minus 4 a squared right parenthesis over denominator left parenthesis 1 minus 4 a squared right parenthesis times left parenthesis 1 minus 4 a squared minus 4 a squared right parenthesis end fraction end cell row blank equals cell negative fraction numerator 4 a square root of 1 minus 4 a squared end root over denominator 1 minus 8 a squared end fraction end cell row blank blank blank end table end style$