6.6.2 Menyelesaikan Persamaan Trigonometri (Pemfaktoran)
Contoh:
Cari semua sudut untuk 0° ≤ x ≤ 360° yang boleh selesaikan setiap persamaan trigonometri yang berikut.
(a) kot x = –2 kos x
(b) 3 sek x = 4 kos x
(c) 16 tan x = kot x
Penyelesaian:
(a)
kot x = –2 kos x
\(\frac{\cos x}{\sin x}=-2\cos x\)
kos x = –2 kos x sin x
kos x + 2 sin x kos x = 0
kos x (1 + 2 sin x ) = 0
kos x = 0
x= 90o, 270o
1 + 2 sin x= 0
sin x = –½
∠asas x = 30o
x = (180o + 30o), (360o – 30o)
x = 210o, 330o
Oleh itu, x = 90o, 210o, 270o, 330o
\(\begin{array}{l}3\mathrm{sek}x=4kosx\\\frac3{kosx}=4kosx\\3=4kos^2x\\kos^2x=\frac34\\kosx=\pm\frac{\sqrt3}2\\\angle\text{asas}=30^\circ\\x=30^\circ,{(180^\circ-30^\circ)},{(180^\circ+30^\circ)},{(360^\circ-30^\circ)}\\x=30^\circ,150^\circ,210^\circ,330^\circ\end{array}\)
(c)
\(\begin{array}{l}16\tan x=kotx\\16\tan x=\frac1{\tan x}\\\tan^2x=\frac1{16}\\\tan x=\pm\frac14\\\angle\text{asas}=14.04^\circ\\x=14.04^\circ,{(180^\circ-14.04^\circ)},\\{(180^\circ+14.04^\circ)},{(360^\circ-14.04^\circ)}\\x=14.04^\circ,165.96^\circ,194.04^\circ,345.96^\circ\end{array}\)