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Tentukan nilai dari: a.

Pertanyaan

Tentukan nilai dari:

a. C o s space 90 degree times C o s space 30 degree plus sin space 90 degree times sin space 30 degree equals...

Pembahasan Soal:

Ingat bahwa:

C o s A space cos space B plus sin space A space sin space B equals cos space open parentheses A minus B close parentheses, sehingga:

table attributes columnalign right center left columnspacing 0px end attributes row cell C o s space 90 degree times C o s space 30 degree plus sin space 90 degree times sin space 30 degree end cell equals cell cos space open parentheses 90 minus 30 close parentheses end cell row blank equals cell cos space 60 end cell row blank equals cell 1 half end cell end table

Jadi, nilai dari C o s space 90 degree times C o s space 30 degree plus sin space 90 degree times sin space 30 degree  adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

M. Nasrullah

Mahasiswa/Alumni Universitas Negeri Makassar

Terakhir diupdate 05 Juni 2021

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Pertanyaan yang serupa

Jika   untuk sudut lancip, nilai  sama dengan...

Pembahasan Soal:

Ingat rumus-rumus trigonometri berikut :

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space left parenthesis straight A plus straight B right parenthesis end cell equals cell cos space straight A space cos space straight B space minus space sin space straight A space sin space straight B end cell row cell cos space 2 straight A end cell equals cell 2 cos squared space straight A minus 1 end cell row cell sin space 2 straight A end cell equals cell 2 space sin space straight A space cos space straight A end cell end table

Berdasarkan rumus di atas, maka

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 3 theta end cell equals cell cos space open parentheses 2 theta plus theta close parentheses end cell row blank equals cell cos space 2 theta space cos space theta minus sin space 2 theta space sin space theta end cell row blank equals cell open parentheses 2 cos squared space theta minus 1 close parentheses space cos space theta minus space 2 space sin space theta space cos theta space sin space theta end cell row blank equals cell 2 cos cubed space theta minus cos space theta minus 2 sin squared space theta space cos space theta end cell end table

Selanjutnya diketahui bahwa tan space theta equals a dan theta sudut lancip, dengan segitiga siku-siku bantu,

Sehingga,

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 3 theta end cell equals cell 2 cos cubed space theta minus cos space theta minus 2 sin squared space theta space cos space theta end cell row blank equals cell 2 open parentheses fraction numerator 1 over denominator square root of 1 plus a squared end root end fraction close parentheses cubed minus fraction numerator 1 over denominator square root of 1 plus a squared end root end fraction minus 2 open parentheses fraction numerator a over denominator square root of 1 plus a squared end root end fraction close parentheses squared open parentheses fraction numerator 1 over denominator square root of 1 plus a squared end root end fraction close parentheses end cell row blank equals cell fraction numerator 2 minus open parentheses 1 plus a squared close parentheses minus 2 a squared over denominator open parentheses 1 plus a squared close parentheses square root of 1 plus a squared end root end fraction end cell row blank equals cell fraction numerator 1 minus 3 a squared over denominator open parentheses 1 plus a squared close parentheses square root of 1 plus a squared end root end fraction cross times fraction numerator square root of 1 plus a squared end root over denominator square root of 1 plus a squared end root end fraction end cell row blank equals cell open parentheses fraction numerator 1 minus 3 a squared over denominator open parentheses 1 plus a squared close parentheses squared end fraction close parentheses square root of 1 plus a squared end root end cell end table end style

Jadi, jawaban yang tepat adalah D.

0

Roboguru

Himpunan penyelesaian persamaan untuk adalah...

Pembahasan Soal:

Ingat rumus dasar persamaan trigonometri dan rumus sudut rangkap berikut:

rightwards double arrow cos space 2 A equals 1 minus 2 sin squared space A rightwards double arrow table attributes columnalign right center left columnspacing 0px end attributes row cell sin space straight A end cell equals cell sin space straight B end cell row blank rightwards arrow cell straight A equals straight B plus straight k times 360 degree end cell end table space space space space space space table attributes columnalign right center left columnspacing 0px end attributes row blank rightwards arrow cell straight A equals left parenthesis 180 degree minus straight B right parenthesis plus straight k times 360 degree end cell end table dengan space straight k space adalah space bilangan space bulat

Berdasarkan rumus d atas, maka

sin space straight x degree minus cos space 2 straight x degree equals 0 sin space straight x degree minus open parentheses 1 minus 2 sin squared space straight x degree close parentheses equals 0 2 sin squared space straight x degree plus sin space straight x degree minus 1 equals 0 open parentheses 2 sin space straight x degree minus 1 close parentheses open parentheses sin space straight x plus 1 close parentheses equals 0 sin space straight x degree equals 1 half space atau space sin space straight x equals negative 1

Selanjutnya gunakan rumus dasar persamaan sinus:

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space x degree end cell equals cell 1 half end cell row cell sin space x degree end cell equals cell space sin space 30 degree end cell row blank rightwards arrow cell straight x equals 30 plus straight k times 360 end cell row blank equals cell 30 space left parenthesis untuk space straight k equals 0 right parenthesis end cell row blank rightwards arrow cell straight x equals open parentheses 180 minus 30 close parentheses plus straight k times 360 end cell row blank equals cell 150 plus straight k times 360 end cell row blank equals cell 150 space left parenthesis untuk space straight k equals 0 right parenthesis end cell end table  table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row cell sin space x degree end cell equals cell negative 1 end cell row cell sin space x degree end cell equals cell space sin space 270 degree end cell row blank rightwards arrow cell straight x equals 270 plus straight k times 360 end cell row blank equals cell 270 space left parenthesis untuk space straight k equals 0 right parenthesis end cell row blank rightwards arrow cell straight x equals open parentheses 180 minus 270 close parentheses plus straight k times 360 end cell row blank equals cell negative 90 plus straight k times 360 end cell row blank equals cell 270 space left parenthesis untuk space straight k equals 1 right parenthesis end cell end table

Dengan demikian, nilai x yang memenuhi pada interval 0 less or equal than x less or equal than 360 adalah open curly brackets 30 semicolon space 150 semicolon space 270 close curly brackets.

Jadi, jawaban yang tepat adalah D.

0

Roboguru

Himpunan penyelesaian  persamaan untuk 0 < x < 360 adalah...

Pembahasan Soal:

C o s space 2 x minus sin space x minus 1 equals 0 space  1 minus 2 sin 2 x minus sin space x minus 1 equals 0 space  minus 2 sin 2 x minus sin x equals 0 space  S i n space x left parenthesis negative 2 space sin space x minus 1 right parenthesis equals 0 space  S i n space x equals space 0 space space  x equals 180 to the power of o space end exponent  minus 2 sin space x minus 1 equals 0 space  minus 2 sin space x equals space 1 space  S i n space x equals negative 1 half  x equals 210 to the power of o comma space 330 to the power of o  s e h i n g g a space h i m p u n a n space p e n y e l e s a i a n space y a n g space m e m e n u h i space y a i t u space left curly bracket 180 comma 210 comma 330 right curly bracket

0

Roboguru

Himpunan penyelesaian persamaan cos 2x - 3 cos x + 2 = 0 untuk 0  x   adalah...

Pembahasan Soal:

Ingat! 

begin mathsize 12px style cos space 2 x equals 2 space cos squared space x minus 1  cos space 2 x space minus space 3 space cos space x space plus space 2 space equals space 0  open parentheses 2 space cos squared space x minus 1 close parentheses minus 3 space cos space x plus 2 equals 0  open parentheses 2 space cos squared space x close parentheses minus space 3 space cos space x plus 1 equals 0 end style

cos x = begin mathsize 12px style 1 half end style atau cos x = 1

Diperoleh:

cos x = begin mathsize 12px style 1 half end style begin mathsize 12px style left right double arrow end style x = begin mathsize 12px style 60 degree comma space 300 degree end style

cos x = 1 begin mathsize 12px style left right double arrow end style x = begin mathsize 12px style 0 degree comma space 360 degree end style

Jadi, himpunan penyelesaiannya adalah begin mathsize 12px style open curly brackets 0 comma space straight pi over 3 comma space 5 over 3 straight pi comma space 2 straight pi close curly brackets end style.

0

Roboguru

Himpunan penyelesaian persamaan adalah....

Pembahasan Soal:

Himpunan penyelesaian persamaan cos space 2 x to the power of 0 space plus space 7 space sin space x to the power of 0 space minus space 4 space equals space 0 comma space 0 less or equal than x less or equal than 360

cos space 2 x plus 7 space sin space x minus 4 equals 0  open parentheses 1 minus 2 sin squared x close parentheses plus 7 space sin space x minus 4 equals 0  minus 2 sin squared x plus 7 space sin space x minus 3 equals 0  2 sin squared x minus 7 space sin space x plus 3 equals 0  left parenthesis 2 sin space x minus 1 right parenthesis left parenthesis sin space x minus 3 right parenthesis equals 0  sin space x equals 1 half  sin space x equals sin space 30  x equals 30 plus k.360  x equals 30 comma 150  sin space x equals 3 left parenthesis t i d a k space b e r l a k u  H P space rightwards arrow open curly brackets 30 to the power of 0 comma 150 to the power of 0 close curly brackets

1

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