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23.
\(tan^2x\ge0\Rightarrow y\le2\)
\(y_{max}=2\) khi \(tanx=0\)
\(y_{min}\) không tồn tại
24.
\(-1\le cosx\le1\Rightarrow0< 1+cosx\le2\)
\(\Rightarrow y\ge\frac{1}{2}\)
\(y_{min}=\frac{1}{2}\) khi \(cosx=1\)
\(y_{max}\) ko tồn tại
19.
\(y=\sqrt{5-\frac{1}{2}\left(2sinxcosx\right)^2}=\sqrt{5-\frac{1}{2}sin^22x}\)
\(0\le sin^22x\le1\Rightarrow\frac{3\sqrt{2}}{2}\le y\le\sqrt{5}\)
\(y_{min}=\frac{3\sqrt{2}}{2}\) khi \(sin^22x=1\)
\(y_{max}=\sqrt{5}\) khi \(sin^22x=0\)
21.
\(y=2sin^2x-\left(1-2sin^2x\right)=4sin^2x-1\)
\(0\le sin^2x\le1\Rightarrow-1\le y\le3\)
\(y_{min}=-1\) khi \(sin^2x=0\)
\(y_{max}=3\) khi \(sin^2x=1\)
a/ ĐKXĐ: \(\left\{{}\begin{matrix}sinx\ne1\\sinx\ne-\frac{1}{2}\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x\ne\frac{\pi}{2}+k2\pi\\x\ne-\frac{\pi}{6}+k2\pi\\x\ne\frac{7\pi}{6}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow cosx-sin2x=\sqrt{3}\left(1+sinx-2sin^2x\right)\)
\(\Leftrightarrow cosx-sin2x=\sqrt{3}\left(cos2x+sinx\right)\)
\(\Leftrightarrow\sqrt{3}sinx-cosx=sin2x+\sqrt{3}cos2x\)
\(\Leftrightarrow\frac{\sqrt{3}}{2}sinx-\frac{1}{2}cosx=\frac{1}{2}sin2x+\frac{\sqrt{3}}{2}cos2x\)
\(\Leftrightarrow sin\left(x-\frac{\pi}{3}\right)=sin\left(2x+\frac{\pi}{6}\right)\)
\(\Leftrightarrow...\)
b/ ĐKXĐ: \(cosx+\sqrt{3}sinx\ne0\Leftrightarrow sin\left(x+\frac{\pi}{6}\right)\ne0\Rightarrow...\)
Đặt \(cosx+\sqrt{3}sinx=2sin\left(x+\frac{\pi}{6}\right)=a\) với \(-2\le a\le2\):
\(a=\frac{3}{a}+1\Leftrightarrow a^2-a-3=0\)
\(\Rightarrow\left[{}\begin{matrix}a=\frac{1+\sqrt{13}}{2}>2\left(l\right)\\a=\frac{1-\sqrt{13}}{2}\end{matrix}\right.\)
\(\Rightarrow2sin\left(x+\frac{\pi}{6}\right)=\frac{1-\sqrt{13}}{2}\)
\(\Rightarrow sin\left(x+\frac{\pi}{6}\right)=\frac{1-\sqrt{13}}{4}=sin\alpha\)
\(\Rightarrow\left[{}\begin{matrix}x+\frac{\pi}{6}=\alpha+k2\pi\\x+\frac{\pi}{6}=\pi-\alpha+k2\pi\end{matrix}\right.\) \(\Rightarrow x=...\)
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: \(x\ne k\pi\)
\(1-sin2x=2sin^2x\)
\(\Leftrightarrow1-2sin^2x-sin2x=0\)
\(\Leftrightarrow cos2x-sin2x=0\)
\(\Leftrightarrow cos\left(2x+\frac{\pi}{4}\right)=0\)
\(\Leftrightarrow...\)
1.
\(\Leftrightarrow\left(sinx+cosx\right)\left(sin^2x+cos^2x-sinx.cosx\right)+sinx.cosx-1=0\)
\(\Leftrightarrow\left(sinx+cosx\right)\left(1-sinx.cosx\right)-\left(1-sinx.cosx\right)=0\)
\(\Leftrightarrow\left(sinx+cosx-1\right)\left(1-sinx.cosx\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sinx+cosx=1\\sinx.cosx=1\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{2}sin\left(x+\frac{\pi}{4}\right)=1\\\frac{1}{2}sin2x=1\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}sin\left(x+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}\\sin2x=2\left(vn\right)\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x+\frac{\pi}{4}=\frac{\pi}{4}+k2\pi\\x+\frac{\pi}{4}=\frac{3\pi}{4}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow...\)
2.
\(\Leftrightarrow\frac{1}{2}cosx+\frac{\sqrt{3}}{2}sinx=cos2x\)
\(\Leftrightarrow cos2x=cos\left(x-\frac{\pi}{3}\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}2x=x-\frac{\pi}{3}+k2\pi\\2x=\frac{\pi}{3}-x+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow...\)
3.
\(\Leftrightarrow\sqrt{3}cosx-3sinx=2sin5x-2sinx\)
\(\Leftrightarrow\sqrt{3}cosx-sinx=2sin5x\)
\(\Leftrightarrow-\left(\frac{1}{2}sinx-\frac{\sqrt{3}}{2}cosx\right)=sin5x\)
\(\Leftrightarrow sin5x=-sin\left(x-\frac{\pi}{3}\right)=sin\left(\frac{\pi}{3}-x\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}5x=\frac{\pi}{3}-x+k2\pi\\5x=\frac{2\pi}{3}+x+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow...\)
7.
\(\Leftrightarrow\left[{}\begin{matrix}2x-40^0=60^0+k360^0\\2x-40^0=120^0+n360^0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=50^0+k180^0\\x=80^0+n180^0\end{matrix}\right.\)
Do \(-180^0\le x\le180^0\Rightarrow\left\{{}\begin{matrix}-180^0\le50^0+k180^0\le180^0\\-180^0\le80^0+n180^0\le180^0\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}-\frac{23}{18}\le k\le\frac{13}{18}\\-\frac{13}{9}\le n\le\frac{5}{9}\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}k=\left\{-1;0\right\}\\n=\left\{-1;0\right\}\end{matrix}\right.\)
\(\Rightarrow x=\left\{-130^0;50^0;-100^0;80^0\right\}\)
8.
\(\Leftrightarrow sinx=-\frac{\sqrt{2}}{2}\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-\frac{\pi}{4}+k2\pi\\x=\frac{5\pi}{4}+k2\pi\end{matrix}\right.\)
5.
\(\Leftrightarrow\frac{\sqrt{2}}{2}sin2x+\frac{\sqrt{2}}{2}cos2x=\frac{\sqrt{2}}{2}\)
\(\Leftrightarrow sin2x.sin\frac{\pi}{4}+cos2x.cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\)
\(\Leftrightarrow sin\left(2x+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}\)
\(\Leftrightarrow\left[{}\begin{matrix}2x+\frac{\pi}{4}=\frac{\pi}{4}+k2\pi\\2x+\frac{\pi}{4}=\frac{3\pi}{4}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=k\pi\\x=\frac{\pi}{4}+k\pi\end{matrix}\right.\)
6.
\(\Leftrightarrow2sin2x=-1\)
\(\Leftrightarrow sin2x=-\frac{1}{2}\)
\(\Leftrightarrow\left[{}\begin{matrix}2x=-\frac{\pi}{6}+k2\pi\\2x=\frac{7\pi}{6}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-\frac{\pi}{12}+k\pi\\x=\frac{7\pi}{12}+k\pi\end{matrix}\right.\)
e/
\(y=5sinx+6cosx-7\)
\(=\sqrt{61}\left(\frac{5}{\sqrt{61}}sinx+\frac{6}{\sqrt{61}}cosx\right)-7\)
\(=\sqrt{61}\left(sinx.cosa+cosx.sina\right)-7\) (với \(a\in\left(0;\pi\right)\) sao cho \(cosa=\frac{5}{\sqrt{61}}\))
\(=\sqrt{61}.sin\left(x+a\right)-7\)
Do \(-1\le sin\left(x+a\right)\le1\Rightarrow7-\sqrt{61}\le y\le7+\sqrt{61}\)
\(y_{min}=7-\sqrt{61}\) khi \(sin\left(x+a\right)=-1\)
\(y_{max}=7+\sqrt{61}\) khi \(sin\left(x+a\right)=1\)
f/
\(y=2\left(\frac{1}{2}sinx+\frac{\sqrt{3}}{2}cosx\right)+3\)
\(=2sin\left(x+\frac{\pi}{3}\right)+3\)
\(\Rightarrow1\le y\le5\)
\(y_{min}=1\) khi \(sin\left(x+\frac{\pi}{3}\right)=-1\)
\(y_{max}=5\) khi \(x+\frac{\pi}{3}=1\)
c/
\(y=2\left(1-cos2x\right)+sin2x+cos2x\)
\(=sin2x-cos2x+2=\sqrt{2}sin\left(2x-\frac{\pi}{4}\right)+2\)
Do \(-1\le sin\left(2x-\frac{\pi}{4}\right)\le1\)
\(\Rightarrow2-\sqrt{2}\le y\le2+\sqrt{2}\)
\(y_{min}=2-\sqrt{2}\) khi \(sin\left(2x-\frac{\pi}{4}\right)=-1\)
\(y_{max}=2+\sqrt{2}\) khi \(sin\left(2x+\frac{\pi}{4}\right)=1\)
d/
\(y=\left(sin^2x+cos^2x\right)^3-3sin^2x.cos^2x\left(sin^2x+cos^2x\right)\)
\(=1-3sin^2x.cos^2x\)
\(=1-\frac{3}{4}sin^22x\)
Mà \(0\le sin^22x\le1\Rightarrow\frac{1}{4}\le y\le1\)
\(y_{min}=\frac{1}{4}\) khi \(sin^22x=1\)
\(y_{max}=1\) khi \(sin2x=0\)
a) Đk: sinx \(\ne\)0<=>x\(\ne\)k\(\Pi\)
pt<=>\(\sqrt{3}\)(1-cos2x)-cosx=0
<=>\(\sqrt{3}\)[1-(2cos2x-1)]-cosx=0
<=>2\(\sqrt{3}\)-2\(\sqrt{3}\)cos2x-cosx=0
<=>\(\left\{{}\begin{matrix}cosx=\dfrac{\sqrt{3}}{2}\\cosx=-\dfrac{2\sqrt{3}}{3}< -1\left(loai\right)\end{matrix}\right.\)
tới đây bạn tự giải cho quen, chứ chép thì thành ra không hiểu gì thì khổ
b)pt<=>2sin2x+2sin2x=1
<=>2sin2x+2sin2x=sin2x+cos2x
<=>4sinx.cosx+sin2x-cos2x=0
Tới đây là dạng của pt đẳng cấp bậc 2, ta thấy cosx=0 không phải là nghiệm của pt nên ta chia cả hai vế của pt cho cos2x:
pt trở thành:
4tanx+tan2x-1=0
<=>\(\left[{}\begin{matrix}tanx=-2+\sqrt{2}\\tanx=-2-\sqrt{5}\end{matrix}\right.\)
<=>\(\left[{}\begin{matrix}x=arctan\left(-2+\sqrt{5}\right)+k\Pi\\x=arctan\left(-2-\sqrt{5}\right)+k\Pi\end{matrix}\right.\)(k thuộc Z)
Chú ý: arctan tương ứng ''SHIFT tan'' (khi thử nghiệm trong máy tính)
c)Đk: cosx\(\ne\)0<=>x\(\ne\)\(\dfrac{\Pi}{2}\)+kpi
pt<=>cos2x+\(\sqrt{3}\)sin2x=1
<=>1-sin2x+\(\sqrt{3}\)sin2x-1=0
<=>(\(\sqrt{3}\)-1)sin2x=0
<=>sinx=0<=>x=k\(\Pi\)(k thuộc Z)
d)
pt<=>\(\sqrt{3}\)sin7x-cos7x=\(\sqrt{2}\)
Khúc này bạn coi SGK trang 35 người ta giả thích rõ ràng rồi
pt<=>\(\dfrac{\sqrt{3}}{2}\)sin7x-\(\dfrac{1}{2}\)cos7x=\(\dfrac{\sqrt{2}}{2}\)
<=>sin(7x-\(\dfrac{\Pi}{3}\))=\(\dfrac{\sqrt{2}}{2}\)
<=>sin(7x-\(\dfrac{\Pi}{3}\))=sin\(\dfrac{\Pi}{4}\)
Tới đây bạn tự giải nhé, giải ra nghiệm rồi kiểm tra xem nghiệm nào thuộc khoảng ( đề cho) rồi kết luận
Câu d) mình nhầm nhé
<=>sin(7x-\(\dfrac{\Pi}{6}\))=\(\dfrac{\sqrt{2}}{2}\) mới đúng sorry
ĐKXĐ:
\(sinx.cosx+2sinx-cosx-2\ge0\)
\(\Leftrightarrow sinx\left(cosx+2\right)-\left(cosx+2\right)\ge0\)
\(\Leftrightarrow\left(sinx-1\right)\left(cosx+2\right)\ge0\)
\(\Leftrightarrow sinx-1\ge0\) (do \(cosx+2>0\) với mọi x)
\(\Rightarrow sinx=1\)
\(\Rightarrow x=\dfrac{\pi}{2}+k2\pi\)