Rozwiązanie równań z sinusami i kosinusami

[chewinggum]

\(\displaystyle{ \sin^{3}x=12\sin^{2}x}\)
\(\displaystyle{ 4\sin^{3}x-4\sin^{2}x+3sinx=3}\)
\(\displaystyle{ 2\sin^{5}x=3\sin^{3}x-sinx}\)
\(\displaystyle{ \cos5x-\cos(x-\frac{\pi}{3})=0}\)
\(\displaystyle{ \cos^{2}x= \cos x \cdot\sin3x}\)

pomocy!

[Althorion]

Zadanie 1.:
\(\displaystyle{ \sin^{3}x=12\sin^{2}x \\ \sin^2 x ( \sin x - 12 ) = 0 \\ \sin x = 0}\)
Zadanie 2.:
\(\displaystyle{ 4\sin^{3}x-4\sin^{2}x+3sinx=3 \\ t = \sin x \wedge t \in \left< -1; 1 \right> \\
4t^3 - 4t^2 + 3t - 3 = 0 \wedge t \in \left< -1; 1 \right> \\
(t-1)(4t^2 + 3) = 0 \wedge t \in \left< -1; 1 \right> \\
t = 1 \\ \sin x = 1}\)
Zadanie 3.:
\(\displaystyle{ 2 \sin^5 x = 3 \sin^3 x - \sin x \\ t = \sin x \wedge t \in \left< -1; 1 \right> \\
2t^5 - 3t^3 + t = 0 \wedge t \in \left< -1; 1 \right> \\
t(2t^4 - 3t^2 + 1) = 0 \wedge t \in \left< -1; 1 \right> \\
(t = 0 \vee 2t^4 - 3t^2 + 1 = 0) \wedge t \in \left< -1; 1 \right> \\
\text{wprowadzić niewiadomą pomocniczą, rozwiązać, wrócić do } t = \sin x}\)
Zadanie 5.:
\(\displaystyle{ \cos^{2}x= \cos x \cdot\sin3x \\ \cos x ( \cos x - \sin 3x ) = 0}\)