Diketahui sudut A berada di kuadran I dan sudut B berada di kuadran II. Jika cos A=43​ dan cos B=−52​, berapakah nilai sin A+sin B?

Pertanyaan

Diketahui sudut A berada di kuadran I dan sudut B berada di kuadran II. Jika cos space straight A equals 3 over 4 dan cos space straight B equals negative 2 over 5, berapakah nilai sin space straight A plus sin space straight B?

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Jawaban

 nilai sin space straight A plus sin space straight B adalah fraction numerator 5 square root of 7 plus 4 square root of 21 over denominator 20 end fraction.

Pembahasan

Diketahui sudut A berada di kuadran I dan cos space straight A equals 3 over 4. Karena cosinus adalah perbandingan antara sisi samping dan sisi miring suatu segitiga siku-siku, maka

table attributes columnalign right center left columnspacing 0px end attributes row depan equals cell square root of miring squared minus sign samping squared end root end cell row blank equals cell square root of 4 squared minus sign 3 squared end root end cell row blank equals cell square root of 16 minus sign 9 end root end cell row blank equals cell square root of 7 end cell end table

sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space straight A end cell equals cell depan over miring end cell row blank equals cell fraction numerator square root of 7 over denominator 4 end fraction end cell end table

Selanjutnya, cos space straight B equals negative 2 over 5. Maka

table attributes columnalign right center left columnspacing 0px end attributes row depan equals cell square root of miring squared minus sign samping squared end root end cell row blank equals cell square root of 5 squared minus sign 2 squared end root end cell row blank equals cell square root of 25 minus sign 4 end root end cell row blank equals cell square root of 21 end cell end table

Karena sudut B berada di kuadran II, maka nilai sinusnya positif, yaitu

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space straight B end cell equals cell depan over miring end cell row blank equals cell fraction numerator square root of 21 over denominator 5 end fraction end cell end table

Dengan demikian,

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space straight A plus sin space straight B end cell equals cell fraction numerator square root of 7 over denominator 4 end fraction plus fraction numerator square root of 21 over denominator 5 end fraction end cell row blank equals cell fraction numerator 5 square root of 7 plus 4 square root of 21 over denominator 20 end fraction end cell end table

Jadi, nilai sin space straight A plus sin space straight B adalah fraction numerator 5 square root of 7 plus 4 square root of 21 over denominator 20 end fraction.

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