Тригонометрия

[Гост]

Да се пресметне стойността на израза sin α+cos α, ако tg α=[tex]\frac{3}{4}[/tex]

[Добромир Глухаров]

[tex]tg\alpha=\frac{3}{4}\Leftrightarrow \frac{sin\alpha}{cos\alpha}=\frac{3}{4}\Leftrightarrow sin\alpha=\frac{3}{4}cos\alpha[/tex]

Знаем, че [tex]sin^2\alpha+cos^2\alpha=1\Rightarrow \(\frac{3}{4}cos\alpha\)^2+cos^2\alpha=1\Rightarrow \frac{9}{16}cos^2\alpha+cos^2\alpha=1\Rightarrow[/tex][tex]\Rightarrow 1\frac{9}{16}cos^2\alpha=1\Rightarrow \frac{25}{16}cos^2\alpha=1\Rightarrow \frac{5}{4}cos\alpha=\pm 1\Rightarrow cos\alpha=\pm\frac{4}{5}[/tex]

[tex]sin\alpha=\frac{3}{4}cos\alpha=\pm\frac{3}{4}\cdot\frac{4}{5}=\pm\frac{3}{5}[/tex]

[tex]sin\alpha+cos\alpha=\pm\(\frac{3}{5}+\frac{4}{5}\)=\pm\frac{7}{5}[/tex]