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Jika α dan β adalah sudut-sudut lancip dengan tan α=247​ dan cotan α=125​, hitunglah: c. cos (21​α+β) .

Pertanyaan

Jika alpha dan beta adalah sudut-sudut lancip dengan tan space alpha equals 7 over 24 dan cotan space alpha equals 5 over 12, hitunglah:

c. cos space open parentheses 1 half alpha plus beta close parentheses .

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Jawaban

nilai dari cos space open parentheses 1 half alpha plus beta close parentheses adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 23 over 130 square root of 2 end cell end table

Pembahasan

Ingat kembali rumus:

sin space A over 2 equals plus-or-minus square root of fraction numerator 1 minus cos space A over denominator 2 end fraction end root cos space A over 2 equals plus-or-minus square root of fraction numerator 1 plus cos space A over denominator 2 end fraction end root   

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 cell cos space straight A end cell equals cell samping over miring end cell row cell tan space straight A end cell equals cell depan over samping end cell row cell cotan space straight A end cell equals cell samping over depan end cell end table

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

Pada soal di atas, terdapat kekeliruan pada penulisan soal, yaitu c o tan space alpha equals 5 over 12 , kita asumsikan, c o tan space beta equals 5 over 12.

Pertama untuk sudut alpha:

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell table row cell tan space straight alpha equals 7 over 24 depan over samping equals 7 over 24 end cell end table end cell row blank blank cell table row cell rightwards double arrow depan equals 7 end cell row cell rightwards double arrow samping equals 24 end cell end table end cell end table

Sehingga dengan menggunakan teorema Pytahgoras:

table attributes columnalign right center left columnspacing 0px end attributes row miring equals cell square root of depan squared plus samping squared end root end cell row blank equals cell square root of 7 squared plus 24 squared end root end cell row blank equals cell square root of 625 end cell row blank equals 25 end table

Sehingga diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space straight alpha end cell equals cell fraction numerator depan space over denominator miring end fraction end cell row blank equals cell 7 over 25 end cell row cell cos space straight alpha end cell equals cell samping over miring end cell row blank equals cell 24 over 25 end cell end table

Untuk sudut beta:

table row cell table attributes columnalign right center left columnspacing 0px end attributes row cell cotan space straight beta end cell equals cell 5 over 12 end cell row cell samping over depan end cell equals cell 5 over 12 end cell end table end cell end table table attributes columnalign left end attributes row cell rightwards double arrow samping equals 5 end cell row cell rightwards double arrow depan equals 12 end cell end table

Sehingga dengan menggunakan teorema Pytahgoras:

table attributes columnalign right center left columnspacing 0px end attributes row miring equals cell square root of samping squared plus depan squared end root end cell row blank equals cell square root of 5 squared plus 12 squared end root end cell row blank equals cell square root of 169 end cell row blank equals 13 end table

Sehingga diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space beta end cell equals cell fraction numerator depan space over denominator miring end fraction end cell row blank equals cell 12 over 13 end cell row cell cos space beta end cell equals cell samping over miring end cell row blank equals cell 5 over 13 end cell end table 

karena diketahui bahwa sudut alpha dan beta adalah sudut-sudut lancip maka sudut alpha over 2 juga sudut lancip dan berada pada kuadran I, sehingga:

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space open parentheses 1 half alpha plus beta close parentheses end cell equals cell cos space 1 half alpha space cos space beta space minus sin space 1 half alpha space sin space beta end cell row blank equals cell square root of fraction numerator 1 plus cos space alpha over denominator 2 end fraction end root cos space beta space minus square root of fraction numerator 1 minus cos space alpha over denominator 2 end fraction end root sin space beta end cell row blank equals cell square root of fraction numerator 1 plus begin display style 24 over 25 end style over denominator 2 end fraction end root times 5 over 13 minus square root of fraction numerator 1 minus begin display style 24 over 25 end style over denominator 2 end fraction end root times 12 over 13 end cell row blank equals cell square root of fraction numerator begin display style fraction numerator 25 plus 24 over denominator 25 end fraction end style over denominator 2 end fraction end root times 5 over 13 minus square root of fraction numerator begin display style fraction numerator 25 minus 24 over denominator 25 end fraction end style over denominator 2 end fraction end root times 12 over 13 end cell row blank equals cell square root of fraction numerator begin display style 49 end style over denominator 2 times 25 end fraction end root times 5 over 13 minus square root of fraction numerator begin display style 1 end style over denominator 2 times 25 end fraction end root times 12 over 13 end cell row blank equals cell 7 over 5 square root of 1 half end root times 5 over 13 minus 1 fifth times square root of 1 half end root times 12 over 13 end cell row blank equals cell 35 over 65 square root of 1 half end root minus 12 over 65 square root of 1 half end root end cell row blank equals cell fraction numerator 35 minus 12 over denominator 65 end fraction square root of 1 half end root end cell row blank equals cell 23 over 65 square root of 1 half end root end cell row blank equals cell 23 over 65 open parentheses 1 half square root of 2 close parentheses end cell row blank equals cell 23 over 130 square root of 2 end cell end table 

Jadi, nilai dari cos space open parentheses 1 half alpha plus beta close parentheses adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 23 over 130 square root of 2 end cell end table

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Pertanyaan serupa

Jika α dan β adalah sudut-sudut lancip dengan tan α=247​ dan cotan α=125​, hitunglah: h. cos (α−21​β)

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