Задание

Составьте алгоритм решения тригонометрического уравнения \(\sin{x}+\cos{x}-1=0\)

\(2\sin{\frac{x}{2}}cos{\frac{x}{2}}+(\cos^2{\frac{x}{2}}-\sin^2{\frac{x}{2}})-(\cos^2{\frac{x}{2}}+\sin^2{\frac{x}{2}})=0\)

\(2\sin{\frac{x}{2}}cos{\frac{x}{2}}+\cos^2{\frac{x}{2}}-\sin^2{\frac{x}{2}}-\cos^2{\frac{x}{2}}-\sin^2{\frac{x}{2}}=0\)

\(2\sin{\frac{x}{2}}cos{\frac{x}{2}}-\sin^2{\frac{x}{2}}-\sin^2{\frac{x}{2}}=0\)

\(2\sin{\frac{x}{2}}cos{\frac{x}{2}}-2\sin^2{\frac{x}{2}}=0\)

\(2\sin{\frac{x}{2}}(cos{\frac{x}{2}}-\sin{\frac{x}{2}})=0\)

\(2\sin{\frac{x}{2}}=0\) \(cos{\frac{x}{2}}-\sin{\frac{x}{2}}=0\)

\(\frac{x_1}{2}={\pi{n}},{n\in Z}\) \({x_1}=2{\pi{n}},{n\in Z}\)

\(\tg{\frac{x}{2}}=1\) \({\frac{x_2}{2}}={\frac{\pi}{4}}+{\pi{k}}, {k\in Z}\) \({{x_2}}={\frac{\pi}{2}}+2{\pi{k}}, {k\in Z}\)

\({x_1}=2{\pi{n}},{n\in Z}\) \({{x_2}}={\frac{\pi}{2}}+2{\pi{k}}, {k\in Z}\)