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NV
6 tháng 6 2020

\(\frac{\pi}{2}< a< \pi\Rightarrow\pi< 2a< 2\pi\)

\(tan2a< 0\) \(\Rightarrow\frac{3\pi}{2}< 2a< 2\pi\Rightarrow cos2a>0\)

\(\Rightarrow cos2a=\frac{1}{\sqrt{1+tan^22a}}=\frac{3}{5}\)

\(tan\left(2a+\frac{\pi}{4}\right)=\frac{tan2a+tan\frac{\pi}{4}}{1-tan2a.tan\frac{\pi}{4}}=\frac{-\frac{4}{3}+1}{1+\frac{4}{3}}=...\)

16 tháng 5 2020

--.--  \(-\pi>-\frac{3}{2}\pi\) mà
Chắc nhầm đề rồi, phải là \(-\pi>a>-\frac{3}{2}\pi\)mới đúng chứ

16 tháng 5 2020

\(-\pi>a>-\frac{3}{2}\pi\Leftrightarrow\pi>a>\frac{1}{2}\pi\)

\(\cos a=-\frac{4}{5}\Rightarrow\sin a=\frac{3}{5}\)

\(\sin2a=2\sin a.\cos a=2.\frac{3}{5}.\frac{-4}{5}=-\frac{24}{25}\)

\(\cos2a=2\cos^2a-1=\frac{7}{25}\)

\(\sin\left(\frac{5\pi}{2}-a\right)=\sin\left(\frac{\pi}{2}-a\right)=\cos a=-\frac{4}{5}\)

\(\sin\left(a+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}.\frac{3}{5}-\frac{4}{5}.\frac{\sqrt{2}}{2}=-\frac{\sqrt{2}}{10}\)

\(\cos\left(a+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}.\frac{-4}{5}-\frac{\sqrt{2}}{2}.\frac{3}{5}=-\frac{7\sqrt{2}}{10}\)

\(\Rightarrow\tan\left(a+\frac{\pi}{4}\right)=\frac{1}{7}\)

\(\cos^2\left(\frac{a}{2}\right)=\frac{1+\cos a}{2}=\frac{1}{10}\Leftrightarrow\left|\cos\frac{a}{2}\right|=\frac{\sqrt{10}}{10}\)

Mà \(\frac{\pi}{2}>\frac{a}{2}>\frac{\pi}{4}\)

\(\Rightarrow\cos\frac{a}{2}=\frac{\sqrt{10}}{10}\)

NV
8 tháng 6 2020

\(\frac{sina+sin3a+sin2a}{cosa+cos3a+cos2a}=\frac{2sin2a.cosa+sin2a}{2cos2a.cosa+cos2a}=\frac{sin2a\left(2cosa+1\right)}{cos2a\left(2cosa+1\right)}=\frac{sin2a}{cos2a}=tan2a\)

\(cos^2\left(a-\frac{\pi}{4}\right)-sin^2\left(a-\frac{\pi}{4}\right)=cos\left(2a-\frac{\pi}{2}\right)\)

\(=cos\left(\frac{\pi}{2}-2a\right)=sin2a\)

NV
16 tháng 5 2020

\(\frac{a}{2}\in\left(\frac{\pi}{2};\frac{3\pi}{4}\right)\Rightarrow tan\frac{a}{2}< 0\) ; \(sin\frac{a}{2}>0;cos\frac{a}{2}< 0\)

Đặt \(tan\frac{a}{2}=x< 0\)

\(\frac{2x}{1-x^2}=3\Leftrightarrow3x^2+2x-3=0\Rightarrow tan\frac{a}{2}=x=\frac{-1-\sqrt{10}}{3}\)

\(tan2a=\frac{2tana}{1-tan^2a}=\frac{6}{1-9}=-\frac{3}{4}\)

\(tan4a=\frac{2tan2a}{1-tan^22a}=-\frac{24}{7}\)

\(cos\frac{a}{2}=-\frac{1}{\sqrt{1+tan^2\frac{a}{2}}}=\) số thật kinh khủng

\(sin\frac{a}{2}=\sqrt{1-cos^2\frac{a}{2}}=...\)

\(sin\left(\frac{a}{2}+\frac{\pi}{2}\right)=\sqrt{2}\left(sin\frac{a}{2}+cos\frac{a}{2}\right)=...\)

NV
20 tháng 4 2020

a/ \(\frac{\pi}{2}\le y\le\pi\Rightarrow cosy< 0\)

\(\Rightarrow cosy=-\sqrt{1-sin^2y}=-\frac{2\sqrt{2}}{3}\)

\(sin2y=2siny.cosy=2.\left(\frac{1}{3}\right).\left(-\frac{2\sqrt{2}}{3}\right)=-\frac{4\sqrt{2}}{9}\)

\(cos\left(\frac{\pi}{3}-y\right)=cos\frac{\pi}{3}cosy+sin\frac{\pi}{3}siny=\frac{\sqrt{3}-2\sqrt{2}}{6}\)

\(tany+5=\frac{siny}{cosy}+5=5-\frac{\sqrt{2}}{4}\)

b/ \(-\frac{\pi}{2}\le a\le9\Rightarrow sina\le0\)

\(\Rightarrow sina=\sqrt{1-cos^2a}=-\frac{4}{5}\)

\(sin2a=2sina.cosa=-\frac{24}{25}\)

\(cos2a=cos^2a-sin^2a=-\frac{7}{25}\)

\(tan2a=\frac{sin2a}{cos2a}=\frac{24}{7}\)

c/ \(\pi\le a\le\frac{3\pi}{2}\Rightarrow\left\{{}\begin{matrix}sina\le0\\cosa\le0\end{matrix}\right.\)

\(\Rightarrow cosa=-\frac{1}{\sqrt{1+tan^2a}}=-\frac{1}{2}\Rightarrow sina=-\frac{\sqrt{3}}{2}\)

\(\Rightarrow sin2a=2sina.cosa=\frac{\sqrt{3}}{2}\)

\(\Rightarrow\left(\sqrt{3}-sin2a\right)sin\frac{2\pi}{3}=\frac{3}{4}\)

NV
10 tháng 4 2019

Câu 1:

\(tan\left(a+\frac{\pi}{4}\right)=1\Rightarrow a+\frac{\pi}{4}=\frac{\pi}{4}+k\pi\Rightarrow a=k\pi\) (\(k\in Z\) )

Do \(\frac{\pi}{2}< a< 2\pi\Rightarrow\frac{\pi}{2}< k\pi< 2\pi\Rightarrow\frac{1}{2}< k< 2\Rightarrow k=1\Rightarrow a=\pi\)

\(\Rightarrow P=cos\left(\pi-\frac{\pi}{6}\right)+sin\pi=-\frac{\sqrt{3}}{2}\)

Câu 2:

\(cot\left(a+\frac{\pi}{3}\right)=-\sqrt{3}=cot\left(-\frac{\pi}{6}\right)\)

\(\Rightarrow a+\frac{\pi}{3}=-\frac{\pi}{6}+k\pi\Rightarrow a=-\frac{\pi}{2}+k\pi\) (\(k\in Z\))

\(\Rightarrow\frac{\pi}{2}< -\frac{\pi}{2}+k\pi< 2\pi\Rightarrow-\pi< k\pi< \frac{5\pi}{2}\)

\(\Rightarrow-1< k< \frac{5}{2}\Rightarrow k=\left\{0;1;2\right\}\Rightarrow a=\left\{-\frac{\pi}{2};\frac{\pi}{2};\frac{3\pi}{2}\right\}\) \(\Rightarrow cosa=0\)

\(\Rightarrow P=sin\left(\pi+\frac{\pi}{6}\right)+0=-sin\frac{\pi}{6}=-\frac{1}{2}\)

NV
10 tháng 4 2019

Vậy đáp án sai

Bạn thay thử \(a=\frac{3\pi}{2}\) vào biểu thức ban đầu coi có đúng \(cot\left(a+\frac{\pi}{3}\right)=-\sqrt{3}\) ko là biết đáp án đúng hay sai liền mà

NV
7 tháng 6 2020

\(a\in\left(\frac{\pi}{2};\pi\right)\Rightarrow cosa< 0\Rightarrow cosa=-\sqrt{1-sin^2a}=-\frac{4}{5}\)

\(A=\frac{sin\left(4\pi-\frac{\pi}{2}-a\right)}{sin\left(a+\frac{\pi}{4}\right)-cosa}=\frac{-sin\left(a+\frac{\pi}{2}\right)}{sin\left(a+\frac{\pi}{4}\right)-cosa}=\frac{-cosa}{sina.cos\frac{\pi}{4}+cosa.sin\frac{\pi}{4}-cosa}\)

\(=\frac{-\frac{4}{5}}{\frac{3}{5}.\frac{\sqrt{2}}{2}-\frac{4}{5}.\frac{\sqrt{2}}{2}-\frac{4}{5}}=...\)

NV
19 tháng 6 2020

\(\frac{3\pi}{2}< a< 2\pi\Rightarrow cosa>0\Rightarrow cosa=\sqrt{1-sin^2a}=\frac{4}{5}\)

\(tana=\frac{sina}{cosa}=-\frac{3}{4}\)

\(sin2a=2sina.cosa=-\frac{24}{25}\)

\(cos2a=2cos^2a-1=\frac{7}{25}\)

\(tan\left(a+\frac{\pi}{4}\right)=\frac{tana+tan\frac{\pi}{4}}{1-tana.tan\frac{\pi}{4}}=\frac{-\frac{3}{4}+1}{1+\frac{3}{4}}=...\)

c sai đề

\(sin\left(a+\frac{\pi}{4}\right)=sina.cos\frac{\pi}{4}+cosa.sin\frac{\pi}{4}=...\)

\(M=\frac{\left(-\frac{3}{5}\right)^2-\left(\frac{7}{25}\right)^2}{-\frac{3}{4}}=...\)

NV
5 tháng 6 2020

\(\frac{\pi}{2}< a< \frac{3\pi}{2}\Rightarrow cosa< 0\Rightarrow cosa=-\sqrt{1-sin^2a}=-\frac{\sqrt{3}}{2}\)

\(A=cosa.cos\frac{4\pi}{3}+sina.sin\frac{4\pi}{3}=-\frac{\sqrt{3}}{2}.\left(-\frac{1}{2}\right)+\frac{1}{2}.\left(-\frac{\sqrt{3}}{2}\right)=0\)

\(B=cos\left(2a+2019.2\pi\right)=cos2a=1-2sin^2a=1-2\left(\frac{1}{2}\right)^2=\frac{1}{2}\)