Đáp án D.

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\(cos\left(\frac{x}{2}+15^0\right)=sinx=cos\left(90^0-x\right)\)

\(\Rightarrow\left[{}\begin{matrix}\frac{x}{2}+15^0=90^0-x+k360^0\\\frac{x}{2}+15^0=x-90^0+k360^0\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=50^0+k240^0\\x=210^0+k720^0\end{matrix}\right.\)

Với \(k=1\Rightarrow x=290^0\)

Bài 2:

\(\Leftrightarrow2sinx+2sinx.cosx-cosx-cos^2x-sin^2x=0\)

\(\Leftrightarrow2sinx+2sinx.cosx-cosx-1=0\)

\(\Leftrightarrow2sinx\left(cosx+1\right)-\left(cosx+1\right)=0\)

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

\(\Rightarrow\left[{}\begin{matrix}sinx=\frac{1}{2}\\cosx=-1\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=\frac{\pi}{6}+k2\pi\\x=\frac{5\pi}{6}+k2\pi\\x=\pi+k2\pi\end{matrix}\right.\) đáp án B

3/ \(y=\frac{sinx+cosx-1}{sinx-cosx+3}\)

\(\Leftrightarrow y.sinx-y.cosx+3y=sinx+cosx-1\)

\(\Leftrightarrow\left(y-1\right)sinx-\left(y+1\right)cosx=-3y-1\)

Theo điều kiện có nghiệm của pt lượng giác bậc nhất:

\(\left(y-1\right)^2+\left(y+1\right)^2\ge\left(-3y-1\right)^2\)

\(\Leftrightarrow7y^2+6y-1\le0\)

\(\Rightarrow-1\le y\le\frac{1}{7}\Rightarrow y_{max}=\frac{1}{7}\)

a) làm tương tự 2 bài mk đã giải nha.

b) \(y=2\cos^2x-2\sqrt{3}\sin x\cos x+1\)

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

Lại có \(-2\le\cos2x+\sqrt{3}\sin2x\le2\) \(\Rightarrow-1\le y\le3\)

c) Vì \(\left\{{}\begin{matrix}0\le\sqrt[4]{\sin x}\le1\\0\le\sqrt{\cos x}\le1\end{matrix}\right.\)

Do đó \(-1\le y\le1\)

a) = = .

b) = = .

c) = = .

d) y' =\(\dfrac{\left(x^2+7x+3\right)'\left(x^2-3x\right)-\left(x^2+7x+3\right)\left(x^2-3x\right)'}{\left(x^2-3x\right)^2}\)=\(\dfrac{\left(2x+7\right)\left(x^2-3x\right)-\left(x^2+7x+3\right)\left(2x-3\right)}{\left(x^2-3x\right)^2}\)=\(\dfrac{-2x^2-6x+9}{\left(x^2-3x\right)^2}\)

d.

\(\sqrt{2}sin\left(x+\frac{\pi}{4}\right)=\sqrt{2}\)

\(\Leftrightarrow sin\left(x+\frac{\pi}{4}\right)=1\)

\(\Leftrightarrow x+\frac{\pi}{4}=\frac{\pi}{2}+k2\pi\)

\(\Leftrightarrow x=\frac{\pi}{4}+k2\pi\)

e.

\(\Leftrightarrow cosx.cos\left(\frac{\pi}{12}\right)-sinx.sin\left(\frac{\pi}{12}\right)=\frac{1}{2}\)

\(\Leftrightarrow cos\left(x+\frac{\pi}{12}\right)=\frac{1}{2}\)

\(\Leftrightarrow\left[{}\begin{matrix}x+\frac{\pi}{12}=\frac{\pi}{3}+k2\pi\\x+\frac{\pi}{12}=-\frac{\pi}{3}+k2\pi\end{matrix}\right.\)

2.a.

ĐKXĐ: ...

\(\sqrt{3}tanx-\frac{6}{tanx}+2\sqrt{3}-3=0\)

\(\Leftrightarrow\sqrt{3}tan^2x+\left(2\sqrt{3}-3\right)tanx-6=0\)

\(\Leftrightarrow\left[{}\begin{matrix}tanx=-2\\tanx=\sqrt{3}\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=arctan\left(-2\right)+k\pi\\x=\frac{\pi}{3}+k\pi\end{matrix}\right.\)

b.

ĐKXĐ: \(x\ne k\pi\)

\(1-sin2x=2sin^2x\)

\(\Leftrightarrow1-2sin^2x-sin2x=0\)

\(\Leftrightarrow cos2x-sin2x=0\)

\(\Leftrightarrow cos\left(2x+\frac{\pi}{4}\right)=0\)

\(\Leftrightarrow...\)

c/

\(\Leftrightarrow\frac{1}{2}cosx-\frac{\sqrt{3}}{2}sinx=cos3x\)

\(\Leftrightarrow cos\left(x+\frac{\pi}{3}\right)=cos3x\)

\(\Rightarrow\left[{}\begin{matrix}x+\frac{\pi}{3}=3x+k2\pi\\x+\frac{\pi}{3}=-3x+k2\pi\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=\frac{\pi}{6}+k\pi\\x=\frac{\pi}{12}+\frac{k\pi}{2}\end{matrix}\right.\)

d/

\(\Leftrightarrow\frac{1}{2}sin3x-\frac{\sqrt{3}}{2}cos3x=sin2x\)

\(\Leftrightarrow sin\left(3x-\frac{\pi}{3}\right)=sin2x\)

\(\Leftrightarrow\left[{}\begin{matrix}3x-\frac{\pi}{3}=2x+k2\pi\\3x-\frac{\pi}{3}=\pi-2x+k2\pi\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=\frac{\pi}{3}+k2\pi\\x=\frac{4\pi}{15}+\frac{k2\pi}{5}\end{matrix}\right.\)

a/

\(\Leftrightarrow\frac{1}{2}sinx+\frac{\sqrt{3}}{2}cosx=sin\left(x+\frac{\pi}{6}\right)\)

\(\Leftrightarrow sin\left(x+\frac{\pi}{3}\right)=sin\left(x+\frac{\pi}{6}\right)\)

\(\Rightarrow x+\frac{\pi}{3}=\pi-x-\frac{\pi}{6}+k2\pi\)

\(\Rightarrow x=\frac{\pi}{4}+k\pi\)

b/

\(\Leftrightarrow\frac{\sqrt{3}}{2}sinx+\frac{1}{2}cosx=sin\frac{\pi}{12}\)

\(\Leftrightarrow sin\left(x+\frac{\pi}{6}\right)=sin\frac{\pi}{12}\)

\(\Rightarrow\left[{}\begin{matrix}x+\frac{\pi}{6}=\frac{\pi}{12}+k2\pi\\x+\frac{\pi}{6}=\frac{11\pi}{12}+k2\pi\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=-\frac{\pi}{12}+k2\pi\\x=\frac{3\pi}{4}+k2\pi\end{matrix}\right.\)

a/ \(y=2\left(\frac{\sqrt{3}}{2}sinx-\frac{1}{2}cosx\right)+5=2sin\left(x-\frac{\pi}{6}\right)+5\)

Do \(-1\le sin\left(x-\frac{\pi}{6}\right)\le1\Rightarrow3\le y\le7\)

b/ \(y=2cos\left(x+\frac{\pi}{6}\right)cos\left(-\frac{\pi}{6}\right)=\sqrt{3}cos\left(x+\frac{\pi}{6}\right)\)

Do \(-1\le cos\left(x+\frac{\pi}{6}\right)\le1\Rightarrow-\sqrt{3}\le y\le\sqrt{3}\)

c/ \(y=2\left(\frac{1}{2}sinx+\frac{\sqrt{3}}{2}cosx\right)+12=2sin\left(x+\frac{\pi}{3}\right)+12\)

Do \(-1\le sin\left(x+\frac{\pi}{3}\right)\le1\Rightarrow10\le y\le14\)