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Pertanyaan

Tentukan semua sudut theta left parenthesis 0 degree less or equal than theta less or equal than 360 degree right parenthesis yang memenuhi masing-masing persamaan trigonometri berikut.

b. 2 space sin space theta equals fraction numerator 3 over denominator cosec squared space theta end fraction minus 1 

S. Yoga

Master Teacher

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Jawaban terverifikasi

Jawaban

 semua sudut theta left parenthesis 0 degree less or equal than theta less or equal than 360 degree right parenthesis yang persamaan trigonometri tersebut adalah theta equals open curly brackets 90 degree semicolon space 199 comma 47 degree semicolon space 340 comma 53 degree close curly brackets.

Pembahasan

Ingat bahwa:

  • sin space x equals sin space alpha rightwards double arrow x equals open parentheses alpha plus left parenthesis 360 k right parenthesis close parentheses degree rightwards double arrow x equals open parentheses left parenthesis 180 minus alpha right parenthesis degree plus left parenthesis 360 k right parenthesis close parentheses degree 

Sehingga,

 table attributes columnalign right center left columnspacing 0px end attributes row cell 2 sin open parentheses theta close parentheses end cell equals cell fraction numerator 3 over denominator csc squared open parentheses theta close parentheses end fraction minus 1 end cell row cell 2 sin open parentheses theta close parentheses minus fraction numerator 3 over denominator csc squared open parentheses theta close parentheses end fraction plus 1 end cell equals 0 row cell fraction numerator 2 sin open parentheses theta close parentheses csc squared open parentheses theta close parentheses minus 3 plus 1 times csc squared open parentheses theta close parentheses over denominator csc squared open parentheses theta close parentheses end fraction end cell equals 0 row cell fraction numerator 2 csc squared open parentheses theta close parentheses sin open parentheses theta close parentheses minus 3 plus csc squared open parentheses theta close parentheses over denominator csc squared open parentheses theta close parentheses end fraction end cell equals 0 row cell 2 csc squared open parentheses theta close parentheses sin open parentheses theta close parentheses minus 3 plus csc squared open parentheses theta close parentheses end cell equals 0 row cell negative 3 plus csc squared open parentheses theta close parentheses plus 2 csc squared open parentheses theta close parentheses fraction numerator 1 over denominator csc open parentheses theta close parentheses end fraction end cell equals 0 row cell negative 3 plus csc squared open parentheses theta close parentheses plus 2 csc open parentheses theta close parentheses end cell equals 0 row cell open parentheses csc open parentheses theta close parentheses minus 1 close parentheses thin space open parentheses csc open parentheses theta close parentheses plus 3 close parentheses end cell equals 0 end table 

csc open parentheses theta close parentheses equals 1 comma thin space csc open parentheses theta close parentheses equals negative 3 fraction numerator 1 over denominator sin open parentheses theta close parentheses end fraction equals 1 comma space fraction numerator 1 over denominator sin open parentheses theta close parentheses end fraction equals negative 3 

table attributes columnalign right center left columnspacing 0px end attributes row blank rightwards arrow cell fraction numerator 1 over denominator sin open parentheses theta close parentheses end fraction equals 1 end cell row 1 equals cell sin open parentheses theta close parentheses end cell row cell sin open parentheses theta close parentheses end cell equals 1 row blank blank blank row blank rightwards arrow cell theta equals 90 degree plus 360 degree n end cell row cell Untuk space n end cell equals 0 row theta equals cell 90 degree end cell end table 

 

table attributes columnalign right center left columnspacing 0px end attributes row blank rightwards arrow cell fraction numerator 1 over denominator sin open parentheses theta close parentheses end fraction equals negative 3 end cell row 1 equals cell negative 3 sin open parentheses theta close parentheses end cell row 1 equals cell negative 3 sin open parentheses theta close parentheses end cell row cell sin open parentheses theta close parentheses end cell equals cell negative 1 third end cell row blank blank blank row blank rightwards arrow cell theta equals sin to the power of negative 1 end exponent open parentheses negative 1 third close parentheses plus 360 degree n end cell row theta equals cell negative 19 comma 47 degree plus 360 degree n end cell row cell Untuk space n end cell equals 1 row theta equals cell 340 comma 53 degree end cell row blank blank blank row blank rightwards arrow cell theta equals 180 degree plus sin to the power of negative 1 end exponent open parentheses 1 third close parentheses plus 360 degree n end cell row theta equals cell 180 degree plus 19 comma 47 degree plus 360 degree n end cell row cell Untuk space n end cell equals 0 row theta equals cell 199 comma 47 degree end cell end table 

Jadi, semua sudut theta left parenthesis 0 degree less or equal than theta less or equal than 360 degree right parenthesis yang persamaan trigonometri tersebut adalah theta equals open curly brackets 90 degree semicolon space 199 comma 47 degree semicolon space 340 comma 53 degree close curly brackets.

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