помогите решить пример. Created by raximovjasur508. matematika-ru

Дано:Найти Решение:1) 2) 3) 4) 5) Ответ:

помогите решить пример...

raximovjasur508 @raximovjasur508

July 2022 1 6 Report

помогите решить пример

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zinaidazina

Дано:

$tg(\alpha+\frac{\pi }{4})=\frac{2}{3}$

Найти $cos2\alpha$

Решение:

1) $tg(\alpha+\frac{\pi }{4})=\frac{tg\frac{\pi}{4}+tg\alpha }{1-tg\frac{\pi}{4}*tg\alpha}=\frac{1+tg\alpha}{1-tg\alpha }$

$tg(\alpha+\frac{\pi }{4})=\frac{1+tg\alpha}{1-tg\alpha }=\frac{2}{3}$

2) $\frac{1+tg\alpha}{1-tg\alpha }=\frac{2}{3}$

$3*({1+tg\alpha)=2*(1-tg\alpha)$

$3tg\alpha+2tg\alpha=-3+2$

$5tg\alpha=-1$

$tg\alpha =-\frac{1}{5}$

3) $\frac{1}{cos^2\alpha } =1+tg^2\alpha$

$\frac{1}{cos^2\alpha } =1+(-\frac{1}{5})^2$

$\frac{1}{cos^2\alpha } =\frac{26}{25}$

$cos^2\alpha =\frac{25}{26}$

4) $sin^2\alpha+cos^2\alpha =1$

$sin^2\alpha =1-cos^2\alpha$

$sin^2\alpha =1-\frac{25}{26}$

$sin^2\alpha =\frac{1}{26}$

5) $cos2\alpha =cos^2\alpha -sin^2\alpha$

$cos2\alpha =\frac{25}{26}-\frac{1}{26}=\frac{24}{26}=\frac{12}{13}$

Ответ: $cos2\alpha =\frac{12}{13}$

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