Câu 4:

Đặt \(x=sina+cosa>0\Rightarrow x^2=\left(sina+cosa\right)^2\)

\(\Rightarrow x^2=sin^2a+cos^2a+2sina.cosa=1+2.\frac{12}{25}=\frac{49}{25}\)

\(\Rightarrow x=\sqrt{\frac{49}{25}}=\frac{7}{5}\)

\(\Rightarrow P=\left(sinx+cosx\right)\left(sin^2x+cos^2x-sinx.cosx\right)\)

\(P=\frac{7}{5}\left(1-\frac{12}{25}\right)=\frac{91}{125}\)

Câu 5:

\(sina+cosa=m\Rightarrow\left(sina+cosa\right)^2=m^2\)

\(\Leftrightarrow sin^2a+cos^2a+2sina.cosa=m^2\)

\(\Leftrightarrow1+2sina.cosa=m^2\)

\(\Rightarrow2sina.cosa=m^2-1\)

\(P=\left|sina-cosa\right|\ge0\)

\(\Leftrightarrow P^2=\left(sina-cosa\right)^2=sin^2a+cos^2a-2sina.cosa\)

\(\Leftrightarrow P^2=1-2sina.cosa=1-\left(m^2-1\right)=2-m^2\)

\(\Rightarrow P=\sqrt{2-m^2}\)

Câu 1:

Do \(\frac{\pi}{2}< a< \pi\Rightarrow cosa< 0\)

\(sin\left(\pi+a\right)=-sina\Rightarrow-sina=-\frac{1}{3}\Rightarrow sina=\frac{1}{3}\)

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

\(P=tan\left(\frac{7\pi}{2}-a\right)=tan\left(3\pi+\frac{\pi}{2}-a\right)=tan\left(\frac{\pi}{2}-a\right)=cota\)

\(\Rightarrow P=\frac{cosa}{sina}=-2\sqrt{2}\)

Câu 2:

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

\(\Rightarrow\frac{tana+1}{1-tana}=1\Rightarrow tana+1=1-tana\Rightarrow tana=0\)

\(\Rightarrow\frac{sina}{cosa}=0\Rightarrow sina=0\)

Do \(\frac{\pi}{2}< a< 2\pi\Rightarrow-1\le cosa< 1\)

\(cos^2a=1-sin^2a=1-0=1\Rightarrow\left[{}\begin{matrix}cosa=-1\\cosa=1\left(l\right)\end{matrix}\right.\)

\(\Rightarrow P=cos\left(a-\frac{\pi}{6}\right)+sina=cosa.cos\frac{\pi}{6}+sina.sin\frac{\pi}{6}+sina\)

\(P=-1.\frac{\sqrt{3}}{2}+0.\frac{1}{3}+0=-\frac{\sqrt{3}}{2}\)

Câu 3:

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

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

\(\Leftrightarrow\frac{1}{2}cosa-\frac{\sqrt{3}}{2}sina+\frac{\sqrt{3}}{2}sina+\frac{3}{2}cosa=0\)

\(\Leftrightarrow2cosa=0\Rightarrow cosa=0\)

Do \(\frac{\pi}{2}< a< 2\pi\Rightarrow-1\le sina< 1\)

\(\Rightarrow sin^2a=1-cos^2a=1\Rightarrow\left[{}\begin{matrix}sina=1\left(l\right)\\sina=-1\end{matrix}\right.\)

\(\Rightarrow P=sin\left(a+\frac{\pi}{6}\right)+cosa=sina.cos\frac{\pi}{6}+cosa.sin\frac{\pi}{6}+cosa\)

\(\Rightarrow P=-\frac{\sqrt{3}}{2}+0+0=-\frac{\sqrt{3}}{2}\)