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Mới thcs mà ế 11 năm (khê lắm)

Cách 1 chắc chắn sai vì pt d' ko cùng phương với d

Còn sai ở đâu thì nhìn cách làm rối loạn quá nên ko biết

Làm cách 1 theo kiểu "cơ bản" thì:

\(A\left(-1;1\right)\Rightarrow\left\{{}\begin{matrix}x_{A'}=-3.\left(-1\right)+\left(1-\left(-3\right)\right).\left(-1\right)=-1\\y_{A'}=-3.1+\left(1-\left(-3\right)\right).3=9\end{matrix}\right.\) \(\Rightarrow A'\left(-1;9\right)\)

\(B\left(2;-1\right)\Rightarrow\left\{{}\begin{matrix}x_{B'}=-3.2+\left(1-\left(-3\right)\right).\left(-1\right)=-10\\y_{B'}=-3.\left(-1\right)+\left(1-\left(-3\right)\right).3=15\end{matrix}\right.\) \(\Rightarrow B'\left(-10;15\right)\)

\(\Rightarrow\overrightarrow{A'B'}=\left(-9;6\right)=3\left(-3;2\right)\)

Phương trình A'B':

\(2\left(x+1\right)+3\left(y-9\right)=0\Leftrightarrow2x+3y-25=0\)

3.3 d)

\(\sin8x-\cos6x=\sqrt{3}\left(\sin6x+\cos8x\right)\\ \Leftrightarrow\sin8x-\sqrt{3}\cos8x=\sqrt{3}\sin6x+\cos6x\\ \Leftrightarrow\sin\left(8x-\dfrac{\pi}{3}\right)=\sin\left(6x+\dfrac{\pi}{6}\right)\\ \Leftrightarrow\left[{}\begin{matrix}8x-\dfrac{\pi}{3}=6x+\dfrac{\pi}{6}+k2\pi\\8x-\dfrac{\pi}{3}=\pi-\left(6x+\dfrac{\pi}{6}\right)+k2\pi\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{4}+k\pi\\x=\dfrac{\pi}{12}+k\dfrac{\pi}{7}\end{matrix}\right.\)

3.4 a)

\(2sin\left(x+\dfrac{\pi}{4}\right)+4sin\left(x-\dfrac{\pi}{4}\right)=\dfrac{3\sqrt{2}}{5}\\ \Leftrightarrow2cos\left(\dfrac{\pi}{2}-x-\dfrac{\pi}{4}\right)+4sin\left(x-\dfrac{\pi}{4}\right)=\dfrac{3\sqrt{2}}{5}\\ \Leftrightarrow2cos\left(-x+\dfrac{\pi}{4}\right)+4sin\left(x-\dfrac{\pi}{4}\right)=\dfrac{3\sqrt{2}}{5}\\ \Leftrightarrow2cos\left(x-\dfrac{\pi}{4}\right)+4sin\left(x-\dfrac{\pi}{4}\right)=\dfrac{3\sqrt{2}}{5}\\ \)

Chia hai vế cho \(\sqrt{2^2+4^2}=2\sqrt{5}\)

Ta được:

\(\dfrac{1}{\sqrt{5}}cos\left(x-\dfrac{\pi}{4}\right)+\dfrac{2}{\sqrt{5}}sin\left(x-\dfrac{\pi}{4}\right)=\dfrac{3}{4}\\ \)

Gọi \(\alpha\) là góc có \(cos\alpha=\dfrac{1}{\sqrt{5}}\)và \(sin\alpha=\dfrac{2}{\sqrt{5}}\)

Phương trình tương đương:

\(cos\left(x-\dfrac{\pi}{4}-\alpha\right)=\dfrac{3}{4}\\ \Leftrightarrow x=\pm arscos\left(\dfrac{3}{4}\right)+\dfrac{\pi}{4}+\alpha+k2\pi\)

Hàm y=cotx là hàm lẻ do cot(-x)=-cotx

Hàm y=sinx là hàm lẻ do sin(-x)=-sinx

Hàm y=cosx là hàm chẵn do cos(-x)=cosx

Hàm y=tanx là hàm lẻ do tan(-x)=-tanx

Câu 1: \(\frac{\pi}{2}<\alpha,\beta<\pi\)

=>\(\sin\alpha>0;\sin\beta>0;cos\alpha<0;cos\beta<0\)

\(\sin^2\alpha+cos^2\alpha=1\)

=>\(cos^2\alpha=1-\sin^2\alpha=1-\left(\frac13\right)^2=\frac89\)

mà \(cos\alpha<0\)

nên \(cos\alpha=-\frac{2\sqrt2}{3}\)

Ta có: \(\sin^2\beta+cos^2\beta=1\)

=>\(\sin^2\beta=1-\left(-\frac23\right)^2=1-\frac49=\frac59\)

mà \(\sin\beta>0\)

nên \(\sin\beta=\frac{\sqrt5}{3}\)

\(\sin\left(\alpha+\beta\right)=\sin\alpha\cdot cos\beta+cos\alpha\cdot\sin\beta\)

\(=\frac13\cdot\frac{-2}{3}+\frac{-2\sqrt2}{3}\cdot\frac{\sqrt5}{3}=\frac{-\sqrt2-2\sqrt{10}}{9}\)

Câu 2:

\(P=cos\left(a+b\right)\cdot cos\left(a-b\right)\)

\(=\frac12\cdot\left\lbrack cos\left(a+b+a-b\right)+cos\left(a+b-a+b\right)\right\rbrack=\frac12\cdot\left\lbrack cos2a+cos2b\right\rbrack\)

\(=\frac12\cdot\left\lbrack2\cdot cos^2a-1+2\cdot cos^2b-1\right\rbrack=cos^2a+cos^2b-1\)

\(=\left(\frac13\right)^2+\left(\frac14\right)^2-1=\frac19+\frac{1}{16}-1=\frac{25}{144}-1=-\frac{119}{144}\)