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Pertanyaan

Jika sin space straight A equals 1 over 10 dan sin space straight B equals 9 over 10, hitunglah :

cos space open parentheses straight A minus straight B close parentheses space text dengan A tumpul dan B lancip end text

A. Salim

Master Teacher

Mahasiswa/Alumni Universitas Pelita Harapan

Jawaban terverifikasi

Jawaban

 cos space open parentheses straight A minus straight B close parentheses equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 3 open parentheses 3 minus square root of 209 close parentheses over denominator 100 end fraction end cell end table.

Pembahasan

Ingat :

1. Rumus pythagoras pada segitiga siku-siku

z squared equals x squared plus y squared

2. Perbandingan trigonometri pada segitiga siku-siku

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space theta end cell equals cell y over z end cell row cell cos space theta end cell equals cell x over z end cell end table

3. Sudut lancip merupakan sudut yang berada pada 0 less than straight pi less than straight pi over 2 dan sudut tumpul merupakan sudut yang berada pada straight pi over 2 less than straight alpha less than straight pi 

4. cos space open parentheses straight pi minus straight alpha close parentheses equals negative cos space alpha

5. Rumus kosinus selisih dua sudut

cos space open parentheses straight A minus straight B close parentheses equals cos space straight A space cos space straight B plus sin space straight A space sin space straight B

Dari soal diketahui  sin space straight A equals 1 over 10 dan sin space straight B equals 9 over 10cos space open parentheses straight A minus straight B close parentheses space text dengan A tumpul dan B lancip end text.

Berdasarkan rumus pythagoras maka :

Dimana A equals straight pi minus straight theta

table attributes columnalign right center left columnspacing 0px end attributes row cell z squared end cell equals cell x squared plus y squared end cell row cell x squared end cell equals cell z squared minus y squared end cell row cell x squared end cell equals cell 10 squared minus 1 squared end cell row cell x squared end cell equals cell 100 minus 1 equals 99 end cell row x equals cell square root of 99 equals 3 square root of 11 end cell end table

Berdasarkan konsep di atas yaitu perbandingan trigonometri pada segitiga siku-siku maka :

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space theta end cell equals cell x over z end cell row cell cos space A end cell equals cell x over z equals fraction numerator 3 square root of 11 over denominator 10 end fraction end cell end table

Maka cos space A equals cos space open parentheses straight pi minus straight theta close parentheses equals negative cos space theta equals negative fraction numerator 3 square root of 11 over denominator 10 end fraction 

Berdasarkan rumus pythagoras maka :

table attributes columnalign right center left columnspacing 0px end attributes row cell z squared end cell equals cell x squared plus y squared end cell row cell x squared end cell equals cell z squared minus y squared end cell row cell x squared end cell equals cell 10 squared minus 9 squared end cell row cell x squared end cell equals cell 100 minus 81 equals 19 end cell row x equals cell square root of 19 end cell end table

Berdasarkan konsep di atas yaitu perbandingan trigonometri pada segitiga siku-siku maka :

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space theta end cell equals cell x over z end cell row cell cos space B end cell equals cell x over z equals 1 over 10 square root of 19 end cell end table

Berdasarkan rumus kosinus selisih dua sudut

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space open parentheses alpha minus beta close parentheses end cell equals cell cos space alpha space cos space beta plus sin space alpha space sin space beta end cell row cell cos space open parentheses straight A minus straight B close parentheses end cell equals cell cos space straight A space cos space straight B plus sin space straight A space sin space straight B end cell row blank equals cell open parentheses fraction numerator negative 3 square root of 11 over denominator 10 end fraction cross times fraction numerator square root of 19 over denominator 10 end fraction close parentheses plus open parentheses 1 over 10 cross times 9 over 10 close parentheses end cell row blank equals cell fraction numerator negative 3 square root of 209 over denominator 100 end fraction plus 9 over 100 equals fraction numerator 3 open parentheses 3 minus square root of 209 close parentheses over denominator 100 end fraction end cell end table

Dengan demikian, cos space open parentheses straight A minus straight B close parentheses equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 3 open parentheses 3 minus square root of 209 close parentheses over denominator 100 end fraction end cell end table.

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