Pertanyaan

Jika 3 tan x + cot x = 5 csc x dan 0 ∘ ≤ x ≤ 36 0 ∘ maka cos x = ....

Jika $3 tan x + cot x = 5 csc x$ dan $0^{\circ} \leq x \leq 36 0^{\circ}$ maka $cos x = ....$

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah A.

Pembahasan

Diketahui , maka berdasarkan definisi tangen yaitu dan cotangen yaitu sertaidentitas trigonometri yaitu sehingga diperoleh Berdasarkan identitas trigonometri dengan maka Misalkan maka 2 cos 2 x + 5 cos x − 3 = 0 2 ( cos x ) 2 + 5 cos x − 3 = 0 2 p 2 + 5 p − 3 = 0 ( 2 p − 1 ) ( p + 3 ) = 0 2 p − 1 = 0 ∨ 2 p = 1 p = 2 1 cos x = 2 1 p + 3 = 0 p = − 3 cos x = − 3 ( tidak memenuhi ) Jadi nilai dari persamaan adalah 2 1 . Oleh karena itu, jawaban yang benar adalah A.

Diketahui 3 tan space x plus cot space x equals 5 csc space x , maka berdasarkan definisi tangen yaitu $tan space x equals fraction numerator sin space x over denominator cos space x end fraction$ dan cotangen yaitu $cot space x equals fraction numerator cos space x over denominator sin space x end fraction$ serta identitas trigonometri yaitu $csc space x equals fraction numerator 1 over denominator sin space x end fraction$ sehingga diperoleh

$table attributes columnalign right center left columnspacing 0px end attributes row cell 3 tan space x plus cot space x end cell equals cell 5 csc space x end cell row cell 3 cross times fraction numerator sin space x over denominator cos space x end fraction plus fraction numerator cos space x over denominator sin space x end fraction end cell equals cell 5 cross times fraction numerator 1 over denominator sin space x end fraction end cell row cell fraction numerator 3 sin space x over denominator cos space x end fraction plus fraction numerator cos space x over denominator sin space x end fraction end cell equals cell fraction numerator 5 over denominator sin space x end fraction end cell row cell fraction numerator 3 sin space x times sin space x plus cos space x times cos space x over denominator sin space x cos space x end fraction end cell equals cell fraction numerator 5 over denominator sin space x end fraction end cell row cell fraction numerator 3 sin squared x plus cos squared x over denominator sin space x cos space x end fraction end cell equals cell fraction numerator 5 over denominator sin space x end fraction end cell row cell 3 sin squared x plus cos squared x end cell equals cell fraction numerator 5 times up diagonal strike sin space x end strike cos space x over denominator up diagonal strike sin space x end strike end fraction end cell row cell 3 sin squared x plus cos squared x end cell equals cell 5 cos space x end cell row cell 3 sin squared x plus cos squared x minus 5 cos space x end cell equals 0 end table$

Berdasarkan identitas trigonometri sin squared x plus cos squared x equals 1 dengan sin squared x equals 1 minus cos squared x maka

Misalkan p equals cos space x maka

$2 cos^{2} x + 5 cos x - 3 = 0 2 (cos x)^{2} + 5 cos x - 3 = 0 2 p^{2} + 5 p - 3 = 0 (2 p - 1) (p + 3) = 0 2 p - 1 = 0 \lor 2 p = 1 p = \frac{1}{2} cos x = \frac{1}{2} p + 3 = 0 p = - 3 cos x = - 3 (tidak memenuhi)$