Trygonometria

[hojrak]

Rozwiąż równania
a) \(\sin 4x-\sin 2x=2\cos 2x-1\) w \([0,2\pi]\)
b) \(\sin x+\sin 2x+\sin 3x=0\) w \([0,\pi]\)
c) \(4\sin 7x\cos 2x=2\sin 9x-1\) w \([0,\pi]\)
d) \(3\cos 2x+10(\cos x)^2=24\sin x-3\), w \([0,2\pi]\)
e) \(2\cos7x-(\cos x)^2=(\sin x)^2+4\cos(2x)\cos(5x)\)

[eresh]

Rozwiąż równania
a) \(sin 4x-sin 2x=2cos 2x-1\) w \([0,2\pi]\)

\(2\sin 2x\cos 2x-\sin 2x-2\cos 2x+1=0\\
-\sin 2x(-2\cos 2x+1)-2\cos 2x+1=0\\
(-2\cos 2x+1)(-\sin 2x+1)=0\\
2\cos 2x=1\;\;\;\vee\;\;\sin 2x=1\\
2x=\frac{\pi}{3}+2k\pi\;\;\;\vee\;\;\;2x=-\frac{\pi}{3}+2k\pi\;\;\;2x=\frac{\pi}{2}+2k\pi,\;\;k\in\mathbb{Z}\\
x=\frac{\pi}{6}+k\pi\;\;\;\vee\;\;\;x=-\frac{\pi}{6}+k\pi\;\;\;x=\frac{\pi}{4}+k\pi\\
x\in\{\frac{\pi}{6},\frac{7\pi}{6},\frac{5\pi}{6},\frac{\pi}{4},\frac{5\pi}{4},\frac{11\pi}{6}\}\)

[eresh]

Rozwiąż równania

b) \(sin x+sin 2x+sin 3x=0\) w \([0,\pi]\)

\(2\sin 2x\cos x+\sin 2x=0\\
\sin 2x(2\cos x+1)+0\\
\sin 2x=0\;\;\;\vee\;\;\cos x=-\frac{1}{2}\\
x=0,5k\pi\;\;\vee\;\;\;x=\frac{2}{3}\pi+2k\pi\;\;\;\vee\;\;\;x=-\frac{2\pi}{3}+2k\pi\\
x\in\{0,\frac{\pi}{2},\pi, \frac{2\pi}{3}\}
\)

[eresh]

Rozwiąż równania

c) \(4sin 7xcos 2x=2sin 9x-1\) w \([0,\pi]\)

\(4\sin\frac{9x+5x}{2}\cos\frac{9x-5x}{2}-2\sin 9x+1=0\\
2\sin 9x+2\sin 5x-2\sin 9x=-1\\
\sin 5x=-\frac{1}{2}\\
5x=-\frac{\pi}{6}+2k\pi\;\;\;\vee\;\;\;5x=-\frac{5\pi}{6}+2k\pi\\
x=-\frac{\pi}{30}+\frac{12k\pi}{30}\;\;\;\vee\;\;\;x=-\frac{-5\pi}{30}+\frac{12k\pi}{30}\\
x\in\{\frac{11\pi}{30},\frac{7\pi}{30},\frac{19\pi}{30},\frac{23\pi}{30}\}\)

[eresh]

Rozwiąż równania

d) \(3cos 2x+10(cos x)^2=24sin x-3\), w \([0,2\pi]\)

\(3-6\sin^2x+10-10\sin ^2x-24\sin x+3=0\\
-16\sin ^2x-24\sin x+16=0\\
\sin x=t, t\in[-1,1]\\
t=\frac{1}{2}\\
\sin x=\frac{1}{2}\\
x\in\{\frac{\pi}{6},\frac{5\pi}{6}\}\)

[eresh]

Rozwiąż równania

e) \(2cos7x-(cosx)^2=(sin x)^2+4cos(2x)cos(5x)\)

\(2\cos 7x-1-4\cos 2x\cos 5x=0\\
2\cos 7x-1-2\cos 7x-2\cos 3x=0\\
\cos 3x=-\frac{1}{2}\\
3x=\frac{2\pi}{3}+2k\pi\;\;\;\vee\;\;\;3x=\frac{4\pi}{3}+2k\pi\\
x=\frac{2\pi}{9}+\frac{2k\pi}{3}\;\;\;\vee\;\;\;x=\frac{4\pi}{9}+\frac{2k\pi}{3}, k\in\mathbb{Z}\)

[kerajs]

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