Pergunakan formula sudut paruh untuk menghitung nilai dari ekspresi di bawah ini: d. sin 105∘ e. cos 105∘ f. tan 105∘

Pertanyaan

Pergunakan formula sudut paruh untuk menghitung nilai dari ekspresi di bawah ini:

d. $\sin\;105^\circ$

e. $\cos\;105^\circ$

f. $\tan\;105^\circ$

M. Nasrullah

Master Teacher

Mahasiswa/Alumni Universitas Negeri Makassar

Jawaban terverifikasi

Pembahasan

Ingat kembali

-rumus sudut paruh:

$sin space 1 half alpha equals plus-or-minus square root of fraction numerator 1 minus cos space alpha over denominator 2 end fraction end root cos space 1 half alpha equals plus-or-minus square root of fraction numerator 1 plus cos space alpha over denominator 2 end fraction end root tan space 1 half alpha equals fraction numerator sin space alpha over denominator 1 plus cos space alpha end fraction$

-dan juga ingat kembali

$\sin\;(180+\alpha)=-\sin\;\alpha\\\cos\;(180+\alpha)=-\cos\;\alpha$

-merasionalkan bentuk akar:

$fraction numerator 1 over denominator a minus square root of b end fraction equals fraction numerator 1 over denominator a minus square root of b end fraction cross times fraction numerator a plus square root of b over denominator a plus square root of b end fraction$

Sehingga diperoleh perhitungan:

e. $\sin\;105^\circ$

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 150 degree end cell equals cell sin space 210 over 2 end cell row blank equals cell plus-or-minus square root of fraction numerator 1 minus cos space 210 over denominator 2 end fraction end root end cell row blank equals cell plus-or-minus square root of fraction numerator 1 minus cos space open parentheses 180 plus 30 close parentheses over denominator 2 end fraction end root end cell row blank equals cell plus-or-minus square root of fraction numerator 1 minus open parentheses negative cos space 30 degree close parentheses over denominator 2 end fraction end root end cell row blank equals cell plus-or-minus square root of fraction numerator 1 plus begin display style fraction numerator square root of 3 over denominator 2 end fraction end style over denominator 2 end fraction end root end cell row blank equals cell plus-or-minus square root of fraction numerator 2 plus square root of 3 over denominator 4 end fraction end root end cell row cell sin space 150 degree end cell equals cell fraction numerator square root of 2 plus square root of 3 end root over denominator 2 end fraction space atau space minus fraction numerator square root of 2 plus square root of 3 end root over denominator 2 end fraction end cell end table$

e. $\cos\;105^\circ$ .

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 150 degree end cell equals cell cos space 210 over 2 end cell row blank equals cell plus-or-minus square root of fraction numerator 1 plus cos space 210 over denominator 2 end fraction end root end cell row blank equals cell plus-or-minus square root of fraction numerator 1 plus cos space open parentheses 180 plus 30 close parentheses over denominator 2 end fraction end root end cell row blank equals cell plus-or-minus square root of fraction numerator 1 plus open parentheses negative cos space 30 degree close parentheses over denominator 2 end fraction end root end cell row blank equals cell plus-or-minus square root of fraction numerator 1 minus begin display style fraction numerator square root of 3 over denominator 2 end fraction end style over denominator 2 end fraction end root end cell row blank equals cell plus-or-minus square root of fraction numerator 2 minus square root of 3 over denominator 4 end fraction end root end cell row cell cos space 150 degree end cell equals cell fraction numerator square root of 2 minus square root of 3 end root over denominator 2 end fraction space atau space minus fraction numerator square root of 2 minus square root of 3 end root over denominator 2 end fraction end cell end table$

e. $\tan\;105^\circ$

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 105 degree end cell equals cell tan space open parentheses fraction numerator 210 degree over denominator 2 end fraction close parentheses end cell row blank equals cell fraction numerator sin space 210 over denominator 1 plus cos space 210 end fraction end cell row blank equals cell fraction numerator sin space open parentheses 180 plus 30 close parentheses over denominator 1 plus cos space open parentheses 180 plus 30 close parentheses end fraction end cell row blank equals cell fraction numerator begin display style negative sin space 30 end style over denominator 1 plus begin display style open parentheses negative cos space 30 close parentheses end style end fraction end cell row blank equals cell fraction numerator negative begin display style 1 half end style over denominator 1 minus begin display style fraction numerator square root of 3 over denominator 2 end fraction end style end fraction end cell row blank equals cell fraction numerator negative begin display style fraction numerator 1 over denominator up diagonal strike 2 end fraction end style over denominator begin display style fraction numerator 2 minus square root of 3 over denominator up diagonal strike 2 end fraction end style end fraction end cell row blank equals cell negative fraction numerator 1 over denominator 2 minus square root of 3 end fraction cross times fraction numerator 2 plus square root of 3 over denominator 2 plus square root of 3 end fraction end cell row blank equals cell fraction numerator negative 2 plus square root of 3 over denominator 4 minus 3 end fraction end cell row blank equals cell negative 2 plus square root of 3 end cell end table$

Dengan demikian, dengan menggunakan formula sudut paruh, maka hasil dari:

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 150 degree end cell equals cell fraction numerator square root of 2 plus square root of 3 end root over denominator 2 end fraction space atau space minus fraction numerator square root of 2 plus square root of 3 end root over denominator 2 end fraction end cell end table$

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 150 degree end cell equals cell fraction numerator square root of 2 minus square root of 3 end root over denominator 2 end fraction space atau space minus fraction numerator square root of 2 minus square root of 3 end root over denominator 2 end fraction end cell end table$

$\tan\;105^\circ=\begin{array}{rcl}&&-2+\sqrt3\end{array}$