równania trygonometryczne

[cytrynka114]

jak oliczyc:

\(\displaystyle{ \sin x= \frac{-3}{5}}\)

\(\displaystyle{ \sin 2x= \frac{1}{2}}\)

jak przekształcić na postać czynnikową równanie:

\(\displaystyle{ 2\cos ^{2}8x-\cos 8x=0}\)

[bb314]

\(\displaystyle{ \ \ \ \ k=\{0,\pm1,\pm2,\ ...\}}\)

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\(\displaystyle{ \sin\frac35 \approx 36,87^o}\)
\(\displaystyle{ \sin x= \frac{-3}{5}\ \ \color{green}\Rightarrow\color{black}\ \ x=180^o+36,87^o+k\cdot 360^o\ \vee\ x=-36,87^o+k\cdot 360^o\ \ \color{green}\Rightarrow\color{black}}\)
\(\displaystyle{ \red x=143.13^o+k\cdot 360^o\ \vee\ x=-36,87^o+k\cdot 360^o}\)

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\(\displaystyle{ \sin30^o=\frac12}\)
\(\displaystyle{ \sin 2x= \frac{1}{2}\ \ \color{green}\Rightarrow\color{black}\ \ 2x=30^o+k\cdot360^o\ \vee\ 2x=180^o-30^o+k\cdot360^o\ \ \color{green}\Rightarrow\color{black}}\)
\(\displaystyle{ \red \ x=15^o+k\cdot180^o\ \vee\ x=75^o+k\cdot180^o}\)

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\(\displaystyle{ 2\cos ^{2}8x-\cos 8x=0\ \ \color{green}\Rightarrow\color{black}\ \ 2\cos8x\left(\cos8x-\frac12\right)=0\ \ \color{green}\Rightarrow\color{black}\ \ \cos8x=0\ \ \vee\ \ \cos8x=\frac12}\)

\(\displaystyle{ \cos8x=0\ \ \color{green}\Rightarrow\color{black}\ \ 8x=\frac{\pi}{2}+k \pi\ \ \color{green}\Rightarrow\color{black}\ \ \red x=\frac{\pi}{16}+\frac{k\pi}{8}\ \ \vee}\)

\(\displaystyle{ \cos8x=\frac12\ \ \color{green}\Rightarrow\color{black}\ \ 8x=\frac{\pi}{3}+2k\pi\ \ \vee\ \ 8x=-\frac{\pi}{3}+2k\pi\ \ \color{green}\Rightarrow\color{black}}\)
\(\displaystyle{ \red \ x=\frac{\pi}{24}+\frac{k\pi}{4}\ \ \vee\ \ x=-\frac{\pi}{24}+\frac{k\pi}{4}}\)

[bartek118]

jak przekształcić na postać czynnikową równanie:

\(\displaystyle{ 2\cos ^{2}8x-\cos 8x=0}\)

Wyciągnąć \(\displaystyle{ \cos 8x}\) przed nawias.