Pertanyaan

Diketahui tan α = 2 1 5 , hitunglah sin α , cos α , sec α , cosec α , dan cotan α !

Diketahui $tan α = \frac{1}{2} 5$ , hitunglah $sin α$ , $cos α$ , $sec α$ , $cosec α$ , dan $cotan α$ !

N. Indah

Master Teacher

Mahasiswa/Alumni Universitas Diponegoro

Jawaban terverifikasi

Jawaban

diperoleh bahwa , , , , dan .

diperoleh bahwa $sin space alpha equals fraction numerator square root of 5 over denominator 3 end fraction$ , cos space alpha equals 2 over 3

, $cos e c space alpha equals fraction numerator 3 square root of 5 over denominator 5 end fraction$ , dan $c o tan space alpha equals fraction numerator 2 square root of 5 over denominator 5 end fraction$ .

Pembahasan

Karena diketahui maka. Diperoleh sisi-sisinya yaitu dan . Sebelum mencari yang ditanyakan, terlebih dahulu kita menentukan panjang sisi miring , yaitu dengan menggunakan teorema phytagoras: sehingga panjang sisi miring adalah . Maka diperoleh: Dengan demikian, diperoleh bahwa , , , , dan .

Karena diketahui maka.

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell tan space straight alpha end cell equals cell Depan over Samping end cell row blank equals cell 1 half square root of 5 end cell row blank equals cell fraction numerator square root of 5 over denominator 2 end fraction end cell end table end style$

Diperoleh sisi-sisinya yaitu text depan end text equals square root of 5 dan text miring end text equals 2 . Sebelum mencari yang ditanyakan, terlebih dahulu kita menentukan panjang sisi miring , yaitu dengan menggunakan teorema phytagoras:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row x equals cell square root of square root of 5 squared plus 2 squared end root end cell row blank equals cell square root of 5 plus 4 end root end cell row blank equals cell square root of 9 end cell row x equals 3 end table end style

sehingga panjang sisi miring adalah text miring end text equals 3 . Maka diperoleh:

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell sin space straight alpha end cell equals cell depan over miring end cell row blank equals cell fraction numerator square root of 5 over denominator 3 end fraction end cell end table end style$

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell cos space straight alpha end cell equals cell samping over miring end cell row blank equals cell 2 over 3 end cell end table end style

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell s e c space straight alpha end cell equals cell 1 over cos end cell row blank equals cell fraction numerator 1 over denominator begin display style 2 over 3 end style end fraction end cell row blank equals cell 3 over 2 end cell row blank blank blank end table end style$

$table attributes columnalign right center left columnspacing 0px end attributes row cell cosec space straight alpha end cell equals cell 1 over sin end cell row blank equals cell fraction numerator 1 over denominator begin display style fraction numerator square root of 5 over denominator 3 end fraction end style end fraction end cell row blank equals cell fraction numerator 3 over denominator square root of 5 end fraction cross times fraction numerator square root of 5 over denominator square root of 5 end fraction end cell row blank equals cell fraction numerator 3 square root of 5 over denominator 5 end fraction end cell end table$

$table attributes columnalign right center left columnspacing 0px end attributes row cell cotan space straight alpha end cell equals cell 1 over tan end cell row blank equals cell fraction numerator 1 over denominator begin display style fraction numerator square root of 5 over denominator 2 end fraction end style end fraction end cell row blank equals cell fraction numerator 2 over denominator square root of 5 end fraction cross times fraction numerator square root of 5 over denominator square root of 5 end fraction end cell row blank equals cell fraction numerator 2 square root of 5 over denominator 5 end fraction end cell end table$

Dengan demikian, diperoleh bahwa $sin space alpha equals fraction numerator square root of 5 over denominator 3 end fraction$ , cos space alpha equals 2 over 3 , s e c space alpha equals 3 over 2 , $cos e c space alpha equals fraction numerator 3 square root of 5 over denominator 5 end fraction$ , dan $c o tan space alpha equals fraction numerator 2 square root of 5 over denominator 5 end fraction$ .