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Sebuah segitiga siku-siku salah satu sudutnya θ∘ dan panjang sisi miringnya R. Keliling segitiga itu adalah 100 cm. a. Tunjukkan bahwa R=(1+sin θ∘+cos θ∘100​) b. Tunjukkan nilai minimum dan maksimum dari R.

Pertanyaan

Sebuah segitiga siku-siku salah satu sudutnya theta degree dan panjang sisi miringnya R. Keliling segitiga itu adalah 100 space cm.

a. Tunjukkan bahwa R equals open parentheses fraction numerator 100 over denominator 1 plus sin space theta degree plus cos space theta degree end fraction close parentheses

b. Tunjukkan nilai minimum dan maksimum dari R.

A. Hadiannur

Master Teacher

Mahasiswa/Alumni Universitas Gadjah Mada

Jawaban terverifikasi

Jawaban

diperoleh nilai  R subscript m i n end subscript equals 100 space open parentheses square root of 2 minus 1 close parentheses space dan space R subscript m a k s end subscript tidak ada.

Pembahasan

Ingat :

  •  

sin space theta equals A over R cos space theta equals B over R

  • a space cos space x plus b space sin space x equals R space cos space open parentheses x minus alpha close parentheses comma space dengan R equals square root of a squared plus b squared end root space dan space alpha equals tan to the power of negative 1 end exponent space open parentheses b over a close parentheses

a. Berdasarkan konsep di atas maka diperoleh :

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space theta end cell equals cell a over R rightwards double arrow a equals R space sin space theta end cell row cell cos space theta end cell equals cell b over R rightwards double arrow b equals R space cos space theta end cell row cell a plus b plus R end cell equals 100 row cell R space sin space theta plus R space cos space theta plus R end cell equals 100 row cell R space open parentheses s in space theta plus cos space theta plus 1 close parentheses end cell equals 100 row R equals cell fraction numerator 100 over denominator 1 plus s in space theta plus cos space theta end fraction end cell end table

Dengan demikian, R equals open parentheses fraction numerator 100 over denominator 1 plus sin space theta degree plus cos space theta degree end fraction close parentheses terbukti.

b. Karena open parentheses 1 comma space 1 close parentheses berada di kuadran pertama maka theta juga di kuadran pertama. Berdasarkan konsep di atas diperoleh :

table attributes columnalign right center left columnspacing 0px end attributes row cell a space cos space x plus b space sin space x end cell equals cell R space cos space open parentheses x minus alpha close parentheses end cell row cell sin space theta plus cos space theta end cell equals cell R space cos space open parentheses theta minus alpha close parentheses comma space space 0 less than theta less than 90 degree end cell row R equals cell square root of a squared plus b squared end root space end cell row R equals cell square root of 1 squared plus 1 squared end root equals square root of 1 plus 1 end root equals square root of 2 end cell row alpha equals cell tan to the power of negative 1 end exponent space open parentheses b over a close parentheses equals tan to the power of negative 1 end exponent space open parentheses 1 over 1 close parentheses equals 45 degree end cell end table

Sehingga diperoleh persamaan square root of 2 space cos space open parentheses theta minus 45 degree close parentheses. Jika theta minus 45 degree naik maka cos space open parentheses theta minus 45 degree close parentheses turun. Diperoleh :

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space open parentheses 90 degree minus 45 degree close parentheses end cell equals cell 1 half square root of 2 less than cos space open parentheses theta minus 45 degree close parentheses less or equal than cos space open parentheses 45 degree minus 45 degree close parentheses equals cos space 0 degree equals 1 end cell row cell 1 half square root of 2 end cell less than cell cos space open parentheses theta minus 45 degree close parentheses less or equal than 1 space space open parentheses kedua space ruas space dikali space square root of 2 close parentheses end cell row cell 1 half cross times 2 end cell less than cell square root of 2 space cos space open parentheses theta minus 45 degree close parentheses less or equal than square root of 2 end cell row 1 less than cell square root of 2 space cos space open parentheses theta minus 45 degree close parentheses less or equal than square root of 2 end cell row 1 less than cell sin space theta space cos space theta less or equal than square root of 2 space space space open parentheses kedua space ruas space ditambah space 1 close parentheses end cell row cell 1 plus 1 end cell less than cell 1 plus sin space theta space cos space theta less or equal than square root of 2 plus 1 end cell row cell fraction numerator 1 over denominator square root of 2 plus 1 end fraction end cell less or equal than cell fraction numerator 1 over denominator 1 plus sin space theta space cos space end fraction less than 1 half space space space space open parentheses kedua space ruas space dikali space 100 close parentheses end cell row cell fraction numerator 100 over denominator square root of 2 plus 1 end fraction end cell less or equal than cell fraction numerator 100 over denominator 1 plus sin space theta space cos space end fraction less than 50 end cell row cell fraction numerator 100 over denominator square root of 2 plus 1 end fraction end cell equals cell fraction numerator 100 over denominator square root of 2 plus 1 end fraction cross times fraction numerator square root of 2 minus 1 over denominator square root of 2 minus 1 end fraction equals fraction numerator 100 space square root of 2 minus 1 over denominator open parentheses square root of 2 plus 1 close parentheses open parentheses square root of 2 minus 1 close parentheses end fraction equals fraction numerator 100 space square root of 2 minus 1 over denominator 2 minus 1 end fraction equals 100 space square root of 2 minus 1 end cell row blank blank blank row cell 100 space square root of 2 minus 1 end cell less or equal than cell fraction numerator 100 over denominator 1 plus sin space theta space cos space end fraction less than 50 end cell row cell 100 space square root of 2 minus 1 end cell less or equal than cell R less than 50 end cell end table

Dengan demikian, diperoleh nilai  R subscript m i n end subscript equals 100 space open parentheses square root of 2 minus 1 close parentheses space dan space R subscript m a k s end subscript tidak ada.

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