-π = -3,14; -2π = -6,28; (-5π)/2 = -7,85.

Vậy (-5π)/2 < -6,32 < -2π.

Do đó điểm M nằm ở góc phần tư thứ II.

Đáp án: B

(h.66) Ta có

A M 2  = MA’ = MA + AA’

Suy ra

Sđ A M 2  = -α + π + k2π, k ∈ Z.

Vậy đáp án là B.

6.13. (h.67) Ta có

Sđ A M 3  = -sđ AM = -α + k2π, k ∈ Z.

Đáp án: D

\(4sin^2x.cosx+2cos2x=cosx+\sqrt{3}sin3x\)

\(\Leftrightarrow2\left(1-cos2x\right).cosx+2cos2x=cosx+\sqrt{3}sin3x\)

\(\Leftrightarrow2cosx-2cos2x.cosx+2cos2x=cosx+\sqrt{3}sin3x\)

\(\Leftrightarrow2cosx-cos3x-cosx+2cos2x=cosx+\sqrt{3}sin3x\)

\(\Leftrightarrow\sqrt{3}sin3x+cos3x=2cos2x\)

\(\Leftrightarrow\dfrac{\sqrt{3}}{2}sin3x+\dfrac{1}{2}cos3x=cos2x\)

\(\Leftrightarrow cos\left(3x-\dfrac{\pi}{3}\right)=cos2x\)

\(\Leftrightarrow3x-\dfrac{\pi}{3}=\pm2x+k2\pi\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{3}+k2\pi\\x=\dfrac{\pi}{15}+\dfrac{k2\pi}{5}\end{matrix}\right.\)

Mysterious Person Akai Haruma Nguyễn Thanh Hằng Mashiro Shiina

1.

Đề là \(x\in\left(0;\frac{\pi}{4}\right)\) hay \(x\in\left[0;\frac{\pi}{4}\right]\) ?

2.

\(sin3x-4sinx.cos2x=0\)

\(\Leftrightarrow sin3x-\left(2sin3x-2sinx\right)=0\)

\(\Leftrightarrow2sinx-sin3x=0\)

\(\Leftrightarrow2sinx-3sinx+4sin^3x=0\)

\(\Leftrightarrow sinx\left(4sin^2x-1\right)=0\)

\(\Leftrightarrow sinx\left(1-2cos2x\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx=0\\cos2x=\frac{1}{2}\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=k\pi\\x=\pm\frac{\pi}{6}+k\pi\end{matrix}\right.\)

3.

\(sin^2x.cosx=0\)

\(\Leftrightarrow sin2x=0\)

\(\Leftrightarrow x=\frac{k\pi}{2}\)

4.

\(\sqrt{3}sin2x+1-cos2x=3\)

\(\Leftrightarrow\frac{\sqrt{3}}{2}sin2x-\frac{1}{2}cos2x=1\)

\(\Leftrightarrow sin\left(2x-\frac{\pi}{6}\right)=1\)

\(\Leftrightarrow2x-\frac{\pi}{6}=\frac{\pi}{2}+k2\pi\)

\(\Leftrightarrow x=\frac{\pi}{3}+k\pi\)

5.

Ko có 4 đáp án thì làm sao biết, có vô số pt tương đương với pt này :)

6.

\(sinx+cosx-2sinx.cosx+1=0\)

Đặt \(sinx+cosx=t\Rightarrow\left\{{}\begin{matrix}\left|t\right|\le\sqrt{2}\\2sinx.cosx=t^2-1\end{matrix}\right.\)

Pt trở thành:

\(t+1-t^2+1=0\)

\(\Leftrightarrow-t^2+t+2=0\)

\(\Leftrightarrow\left[{}\begin{matrix}t=-1\\t=2\left(l\right)\end{matrix}\right.\)

\(\Rightarrow2sinx.cosx=t^2-1=0\)

\(\Leftrightarrow sin2x=0\)

\(\Leftrightarrow x=\frac{k\pi}{2}\)

ĐK:..........

Bình phương 2 vế ta được

\(2-3x+2\sqrt{\left(1-2x\right)\left(1-x\right)}=x+4\)

\(\Leftrightarrow2\sqrt{\left(1-2x\right)\left(1-x\right)}=4x+2\)

\(\Leftrightarrow\sqrt{\left(1-2x\right)\left(1-x\right)}=2x+1\)

\(\Leftrightarrow\left(1-2x\right)\left(1-x\right)=4x^2+4x+1\)

\(\Leftrightarrow1-3x+2x^2=4x^2+4x+1\)

\(\Leftrightarrow2x^2+7x=0\)

\(\Leftrightarrow x\left(2x+7\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}x=0\\x=\frac{-7}{2}\end{cases}}\)

Vậy.........................................

ĐKXĐ:...

\(\Leftrightarrow\frac{36}{\sqrt{x-2}}+4\sqrt{x-2}+\frac{4}{\sqrt{y-1}}+\sqrt{y-1}=28\)

Ta có:

\(VT\ge2\sqrt{\frac{36.4\sqrt{x-2}}{\sqrt{x-2}}}+2\sqrt{\frac{4\sqrt{y-1}}{\sqrt{y-1}}}=28\)

Dấu "=" xảy ra khi và chỉ khi:

\(\left\{{}\begin{matrix}\frac{9}{\sqrt{x-2}}=\sqrt{x-2}\\\frac{4}{\sqrt{y-1}}=\sqrt{y-1}\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=11\\y=5\end{matrix}\right.\)

Đặt \(\left\{{}\begin{matrix}\sqrt{x-2}=a\left(a>0\right)\\\sqrt{y-1}=b\left(b>0\right)\end{matrix}\right.\)

\(\Rightarrow\dfrac{36}{a}+\dfrac{4}{b}=28-4a-b\)

\(\Leftrightarrow\left(\dfrac{36}{a}+4a\right)+\left(\dfrac{4}{b}+b\right)=28\)

\(VT\ge2\sqrt{\dfrac{36}{a}\times4a}+2\sqrt{\dfrac{4}{b}\times b}=28\)

Dấu "=" xảy ra khi \(\left\{{}\begin{matrix}\dfrac{36}{a}=4a\\\dfrac{4}{b}=b\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}a=3\\b=2\end{matrix}\right.\) \(\left(a,b>0\right)\)

\(\Rightarrow\left\{{}\begin{matrix}\sqrt{x-2}=3\\\sqrt{y-1}=2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x=11\\y=5\end{matrix}\right.\) (n)

Vậy . . . >3<