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1.
\(\Leftrightarrow4sinx.cosx+3\left(sinx-cosx\right)=0\)
Đặt \(sinx-cosx=t\Rightarrow\left\{{}\begin{matrix}\left|t\right|\le\sqrt{2}\\2sinx.cosx=1-t^2\end{matrix}\right.\)
Pt trở thành:
\(2\left(1-t^2\right)+3t=0\)
\(\Leftrightarrow-2t^2+3t+2=0\Rightarrow\left[{}\begin{matrix}t=2\left(l\right)\\t=-\frac{1}{2}\end{matrix}\right.\)
\(\Leftrightarrow sinx-cosx=-\frac{1}{2}\)
\(\Leftrightarrow\sqrt{2}sin\left(x-\frac{\pi}{4}\right)=-\frac{1}{2}\)
\(\Leftrightarrow sin\left(x-\frac{\pi}{4}\right)=-\frac{1}{2\sqrt{2}}\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{4}+arcsin\left(-\frac{1}{2\sqrt{2}}\right)+k2\pi\\x=\frac{5\pi}{4}-arcsin\left(-\frac{1}{2\sqrt{2}}\right)+k2\pi\end{matrix}\right.\)
2.
Đặt \(sinx-cosx=t\Rightarrow\left\{{}\begin{matrix}\left|t\right|\le\sqrt{2}\\sin2x=2sinx.cosx=1-t^2\end{matrix}\right.\)
Pt trở thành:
\(1-t^2-4t=4\)
\(\Leftrightarrow t^2+4t+3=0\Rightarrow\left[{}\begin{matrix}t=-1\\t=-3\left(l\right)\end{matrix}\right.\)
\(\Rightarrow sinx-cosx=-1\)
\(\Leftrightarrow\sqrt{2}sin\left(x-\frac{\pi}{4}\right)=-1\)
\(\Leftrightarrow sin\left(x-\frac{\pi}{4}\right)=-\frac{\sqrt{2}}{2}\)
\(\Leftrightarrow\left[{}\begin{matrix}x-\frac{\pi}{4}=-\frac{\pi}{4}+k2\pi\\x-\frac{\pi}{4}=\frac{5\pi}{4}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=k2\pi\\x=\frac{3\pi}{2}+k2\pi\end{matrix}\right.\)
\(cos\left(\frac{x}{2}+15^0\right)=sinx=cos\left(90^0-x\right)\)
\(\Rightarrow\left[{}\begin{matrix}\frac{x}{2}+15^0=90^0-x+k360^0\\\frac{x}{2}+15^0=x-90^0+k360^0\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=50^0+k240^0\\x=210^0+k720^0\end{matrix}\right.\)
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:
\(\left(y-1\right)^2+\left(y+1\right)^2\ge\left(-3y-1\right)^2\)
\(\Leftrightarrow7y^2+6y-1\le0\)
\(\Rightarrow-1\le y\le\frac{1}{7}\Rightarrow y_{max}=\frac{1}{7}\)
y = \(\dfrac{sin^2x}{cosx\left(sinx-cosx\right)}+\dfrac{1}{4}\)
y = \(\dfrac{sin^2x}{sinx.cosx-cos^2x}+\dfrac{1}{4}=\dfrac{\dfrac{sin^2x}{cos^2x}}{\dfrac{sinx.cosx}{cos^2x}-1}+\dfrac{1}{4}\)
y = \(\dfrac{tan^2x}{tanx-1}+\dfrac{1}{4}\)
y = \(\dfrac{4tan^2x+tanx-1}{4tanx-4}\). Đặt t = tanx. Do x ∈ \(\left(\dfrac{\pi}{4};\dfrac{\pi}{2}\right)\) nên t ∈ (1 ; +\(\infty\))\
Ta đươc hàm số f(t) = \(\dfrac{4t^2+t-1}{4t-4}\)
⇒ ymin = \(\dfrac{17}{4}\) khi t = 2. hay x = arctan(2) + kπ
a) √2 cos(x - π/4)
= √2.(cosx.cos π/4 + sinx.sin π/4)
= √2.(√2/2.cosx + √2/2.sinx)
= √2.√2/2.cosx + √2.√2/2.sinx
= cosx + sinx (đpcm)
b) √2.sin(x - π/4)
= √2.(sinx.cos π/4 - sin π/4.cosx )
= √2.(√2/2.sinx - √2/2.cosx )
= √2.√2/2.sinx - √2.√2/2.cosx
= sinx – cosx (đpcm).
1.
\(\Leftrightarrow\frac{\sqrt{3}}{2}sinx-\frac{1}{2}cosx+\frac{\sqrt{2}}{2}=0\)
\(\Leftrightarrow sin\left(x-\frac{\pi}{6}\right)+\frac{\sqrt{2}}{2}=0\)
\(\Leftrightarrow sin\left(x-\frac{\pi}{6}\right)=-\frac{\sqrt{2}}{2}\)
\(\Leftrightarrow\left[{}\begin{matrix}x-\frac{\pi}{6}=-\frac{\pi}{4}+k2\pi\\x-\frac{\pi}{6}=\frac{5\pi}{4}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-\frac{\pi}{12}+k2\pi\\x=\frac{17\pi}{12}+k2\pi\end{matrix}\right.\)
2.
\(\Leftrightarrow\frac{3}{\sqrt{13}}sin2x+\frac{2}{\sqrt{13}}cos2x=\frac{3}{\sqrt{13}}\)
Đặt \(\frac{3}{\sqrt{13}}=cosa\) với \(a\in\left(0;\pi\right)\)
\(\Rightarrow sin2x.cosa+cos2x.sina=cosa\)
\(\Leftrightarrow sin\left(2x+a\right)=sin\left(\frac{\pi}{2}-a\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}2x+a=\frac{\pi}{2}-a+k2\pi\\2x+a=\frac{\pi}{2}+a+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{4}-a+k\pi\\x=\frac{\pi}{4}+k\pi\end{matrix}\right.\)
3.
\(\Leftrightarrow sinx-\sqrt{3}cosx=\sqrt{2}\)
\(\Leftrightarrow\frac{1}{2}sinx-\frac{\sqrt{3}}{2}cosx=\frac{\sqrt{2}}{2}\)
\(\Leftrightarrow sin\left(x-\frac{\pi}{3}\right)=\frac{\sqrt{2}}{2}\)
\(\Leftrightarrow\left[{}\begin{matrix}x-\frac{\pi}{3}=\frac{\pi}{4}+k2\pi\\x-\frac{\pi}{3}=\frac{3\pi}{4}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{7\pi}{12}+k2\pi\\x=\frac{13\pi}{12}+k2\pi\end{matrix}\right.\)
4.
Câu này giống hệt câu a
sin π/6 = 1/2; cos π/6 = √3/2
sin π/4 = √2/2; cos π/4 = √2/2
sin 1,5 = 0,9975; cos 1,5 = 0,0707
sin 2 = 0,9093; cos 2 = -0,4161
sin 3,1 = 0,0416; cos 3,1 = -0,9991
sin 4,25 = -0,8950; cos 4,25 = -0,4461
sin 5 = -0,9589; cos 5 = 0,2837