a) P = cos(\(\frac{\Pi}{2}\) + x) + cos(2π - x) + cos(3π + x)   = -sinx + cosx - cosx = -sinx

a/ Bạn coi lại đề bài, pt này có 1 nghiệm rất xấu ko giải được:

\(\Leftrightarrow1-sin^2x-2\sqrt{3}sinx.cosx=sin^3x+1\)

\(\Leftrightarrow sin^3x+sin^2x+2\sqrt{3}sinx.cosx=0\)

\(\Leftrightarrow sinx\left(sin^2x+sinx+2\sqrt{3}cosx\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx=0\Rightarrow x=k\pi\\sin^2x+sinx+2\sqrt{3}cosx=0\left(1\right)\end{matrix}\right.\)

\(\left(1\right)\Leftrightarrow sin^2x+sinx=-2\sqrt{3}cosx\) (\(cosx\le0\))

\(\Leftrightarrow sin^2x\left(sinx+1\right)^2=12cos^2x\)

\(\Leftrightarrow sin^2x\left(sinx+1\right)^2=12\left(1-sinx\right)\left(1+sinx\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}1+sinx=0\left(2\right)\\sin^2x\left(sinx+1\right)=12\left(1-sinx\right)\left(3\right)\end{matrix}\right.\)

\(\left(2\right)\Leftrightarrow x=-\frac{\pi}{2}+k2\pi\) (thỏa mãn)

\(\left(3\right)\Leftrightarrow sin^3x+sin^2x+12sinx-12=0\)

Pt bậc 3 này có nghiệm thực thuộc \(\left(-1;1\right)\) nhưng rất xấu

b/

\(\Leftrightarrow\frac{3}{5}sin2x+\frac{4}{5}cos2x=-cos2003x\)

Đặt \(\frac{3}{5}=cosa\) với \(a\in\left(0;\pi\right)\)

\(\Rightarrow sin2x.cosa+cos2x.sina=-cos2003x\)

\(\Leftrightarrow sin\left(2x+a\right)=sin\left(2003x-\frac{\pi}{2}\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}2003x-\frac{\pi}{2}=2x+a+k2\pi\\2003x-\frac{\pi}{2}=\pi-2x-a+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{4002}+\frac{a}{2001}+\frac{k2\pi}{2001}\\x=\frac{3\pi}{4010}-\frac{a}{2005}+\frac{k2\pi}{2005}\end{matrix}\right.\)

c/

\(\Leftrightarrow\sqrt{3}sin\left(x-\frac{\pi}{3}\right)+cos\left(\frac{\pi}{3}-x\right)=2sin1972x\)

\(\Leftrightarrow\frac{\sqrt{3}}{2}sin\left(x-\frac{\pi}{3}\right)+\frac{1}{2}cos\left(x-\frac{\pi}{3}\right)=sin1972x\)

\(\Leftrightarrow sin\left(x-\frac{\pi}{3}+\frac{\pi}{6}\right)=sin1972x\)

\(\Leftrightarrow sin\left(x-\frac{\pi}{6}\right)=sin1972x\)

\(\Leftrightarrow\left[{}\begin{matrix}1972x=x-\frac{\pi}{6}+k2\pi\\1972x=\frac{7\pi}{6}-x+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=-\frac{\pi}{11826}+\frac{k2\pi}{1971}\\x=\frac{7\pi}{11838}+\frac{k2\pi}{1973}\end{matrix}\right.\)

\(A=2cosx-3cosx-sin\left(3\pi+\frac{\pi}{2}-x\right)+tan\left(\pi+\frac{\pi}{2}-x\right)\)

\(A=-cosx+sin\left(\frac{\pi}{2}-x\right)+tan\left(\frac{\pi}{2}-x\right)\)

\(A=-cosx+cosx+cotx=cotx\)

\(B=2cosx+sin\left(4\pi+\pi-x\right)+sin\left(2\pi-\frac{\pi}{2}+x\right)-sinx\)

\(B=2cosx+sin\left(\pi-x\right)+sin\left(-\frac{\pi}{2}+x\right)-sinx\)

\(B=2cosx+sinx-sin\left(\frac{\pi}{2}-x\right)-sinx\)

\(B=2cosx-cosx=cosx\)

Nhìn đề bài hãi quá :(

a/ \(A=3\sin\left(5.2\pi+\pi-x\right).\sin\left(2\pi+\frac{\pi}{2}-x\right)+2\sin\left(4.2\pi+\pi+x\right)\)

\(A=3\sin\left(\pi-x\right).\sin\left(\frac{\pi}{2}-x\right)+2\sin\left(\pi+x\right)\)

\(A=3\sin x.\cos x-2\sin x=\sin x\left(3\cos x-2\right)\)

b/ \(B=\sin\left(5.2.180^0+180^0+x\right)-\cos\left(90^0-x\right)+\tan\left(90^0+180^0-x\right)+\cot\left(2.180^0-x\right)\)

\(B=\sin\left(180^0+x\right)-\sin x+\tan\left(90^0-x\right)+\cot\left(-x\right)\)

\(B=-\sin x-\sin x+\cot x-\cot x=-2\sin x\)

c/ \(C=-2\sin\left(-(2\pi+\frac{\pi}{2}-x)\right)-3\cos\left(2\pi+\pi-x\right)+5\sin\left(2.2\pi-\left(\frac{\pi}{2}+x\right)\right)+\cot\left(\pi+\frac{\pi}{2}-x\right)\)

\(C=2\sin\left(\frac{\pi}{2}-x\right)-3\cos\left(\pi-x\right)-5\sin\left(\frac{\pi}{2}+x\right)+\cot\left(\frac{\pi}{2}-x\right)\)

\(2\cos x+3\cos x-5\cos x+\tan x=\tan x\)

d/ \(D=\tan\left(-\left(\pi-x\right)\right).\cos\left(-\left(\frac{\pi}{2}-x\right)\right).\left(-\cos x\right)\)

\(D=\tan\left(\pi-x\right).\cos\left(\frac{\pi}{2}-x\right).\cos x\)

\(D=-\tan x.\sin x.\cos x=-\sin^2x\)

e/ \(E=\cos\left(28.2\pi+\pi+\frac{\pi}{2}-x\right)+\sin\left(-\left(58.2\pi+\pi+\frac{\pi}{2}-x\right)\right)+\cos\left(-\left(46.2\pi+\pi+\frac{\pi}{2}-x\right)\right)+\sin\left(35.2\pi+\pi+\frac{\pi}{2}-x\right)\)

\(E=-\cos\left(\frac{\pi}{2}-x\right)+\sin\left(\frac{\pi}{2}-x\right)-\cos\left(\frac{\pi}{2}-x\right)-\sin\left(\frac{\pi}{2}-x\right)\)

\(E=-2\sin x\)

Thôi, stop ở đây, làm nữa chắc tẩu hỏa nhập ma quá :(

Mình thấy hầu hết các bài này đều có chung 1 điểm, và chắc đó cũng là điểm mà bạn thắc mắc: Đó chính là tách các hạng tử ra và biến đổi

Tách cũng đơn giản thôi, cứ gặp sin, cos thì tách sao cho về dạng 2pi+..., gặp tan, cot thì pi.

Còn tách mấy cái phân số như vầy:

Ví dụ \(\frac{7\pi}{2}\) , 7 chia 2 được 3, ta lấy \(\frac{7}{2}-3=\frac{1}{2}\) thì suy ra: \(\frac{7\pi}{2}=3\pi+\frac{\pi}{2}\)

Đó, thế là được :D

\(2sin\left(\frac{\pi}{2}+x\right)+sin\left(3\pi-x\right)+sin\left(\frac{3\pi}{2}+x\right)+cos\left(\frac{\pi}{2}+x\right)\)

\(=2cosx+sinx-cosx-sinx\)

\(=cosx\)

\(A=cosa\left(sinb.cosc-cosb.sinc\right)+cosb\left(sinc.cosa-cosc.sina\right)+cosc\left(sinacosb-cosasinb\right)\)

\(A=cosasinbcosc-cosacosbsinc+cosacosbsinc-sinacosbcosc+sinacosbcosc-cosasinbcosc\)

\(A=0\)

\(B=sin^2x+\frac{1}{2}\left(cos\frac{2\pi}{3}+cos2x\right)\)

\(B=\frac{1}{2}-\frac{1}{2}cos2x-\frac{1}{4}+\frac{1}{2}cos2x\)

\(B=\frac{1}{4}\)

\(C=\frac{1}{2}-\frac{1}{2}cos2x+\frac{1}{2}-\frac{1}{2}cos\left(\frac{4\pi}{3}+2x\right)+\frac{1}{2}-\frac{1}{2}cos\left(\frac{4\pi}{3}-2x\right)\)

\(C=\frac{3}{2}-\frac{1}{2}cos2x-\frac{1}{2}\left(cos\left(\frac{4\pi}{3}+2x\right)+cos\left(\frac{4\pi}{3}-2x\right)\right)\)

\(C=\frac{3}{2}-\frac{1}{2}cos2x-cos\frac{4\pi}{3}.cos2x\)

\(C=\frac{3}{2}-\frac{1}{2}cos2x+\frac{1}{2}cos2x\)

\(C=\frac{3}{2}\)

\(D=\frac{1}{2}\left[\sqrt{2}sin\left(\frac{\pi}{4}+x\right)\right]^2-sin^2x-sinx.\sqrt{2}cos\left(\frac{\pi}{4}+x\right)\)

\(D=\frac{1}{2}\left(sinx+cosx\right)^2-sin^2x-sinx\left(sinx-cosx\right)\)

\(D=\frac{1}{2}\left(1+2sinx.cosx\right)-sin^2x-sin^2x+sinx.cosx\)

\(D=\frac{1}{2}+sinxcosx+sinxcosx=\frac{1}{2}+sin2x\)

Góc độ cao của thang dựa vào tường là 60º và chân thang cách tường 4,6 m. Chiều dài của thang là

\(A=cos\left(6\pi+\pi-x\right)+sin\left(2\pi+\frac{\pi}{2}-x\right)+tan^2\left(\pi+\frac{\pi}{2}-x\right)-\frac{1}{sin^2\left(7\pi+\pi+x\right)}\)

\(=cos\left(\pi-x\right)+sin\left(\frac{\pi}{2}-x\right)+tan^2\left(\frac{\pi}{2}-x\right)-\frac{1}{sin^2\left(\pi+x\right)}\)

\(=-cosx+cosx+cot^2x-\frac{1}{sin^2x}\)

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

\(=cos\left(4\pi+\pi+x\right)+sin\left(4\pi+\frac{\pi}{2}-x\right)-tan\left(\pi+\frac{\pi}{2}+x\right).cot\left(\pi+\frac{\pi}{2}-x\right)\)

\(=cos\left(\pi+x\right)+sin\left(\frac{\pi}{2}-x\right)-tan\left(\frac{\pi}{2}+x\right).cot\left(\frac{\pi}{2}-x\right)\)

\(=-cosx+cosx-\left(-cotx\right).tanx\)

\(=1\)

Câu 2 bạn coi lại đề

3.

\(1+2sinx.cosx-2cosx+\sqrt{2}sinx+2cosx\left(1-cosx\right)=0\)

\(\Leftrightarrow sin2x-\left(2cos^2x-1\right)+\sqrt{2}sinx=0\)

\(\Leftrightarrow sin2x-cos2x=-\sqrt{2}sinx\)

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

\(\Leftrightarrow sin\left(2x-\frac{\pi}{4}\right)=sin\left(-x\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}2x-\frac{\pi}{4}=-x+k2\pi\\2x-\frac{\pi}{4}=\pi+x+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow...\)

4.

Bạn coi lại đề, xuất hiện 2 số hạng \(cos4x\) ở vế trái nên chắc là bạn ghi nhầm

5.

\(\Leftrightarrow sinx.sin2x-cosx.sin^22x=2cos^2\left(\frac{\pi}{4}-x\right)-1\)

\(\Leftrightarrow sinx.sin2x-cosx.sin^22x=cos\left(\frac{\pi}{2}-2x\right)\)

\(\Leftrightarrow sinx.sin2x-cosx.sin^22x=sin2x\)

\(\Leftrightarrow sin2x\left(sinx-cosx.sin2x-1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sin2x=0\Leftrightarrow x=...\\sinx-cosx.sin2x-1=0\left(1\right)\end{matrix}\right.\)

Xét (1):

\(\Leftrightarrow sinx-1-2sinx.cos^2x=0\)

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

\(\Leftrightarrow2sin^3x-sinx-1=0\)

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

\(\Leftrightarrow...\)

6.

\(sinx.sin4x=\sqrt{2}cos\left(\frac{\pi}{6}-x\right)-2\sqrt{3}cosx.sin2x.cos2x\)

\(\Leftrightarrow sinx.sin4x=\sqrt{2}cos\left(\frac{\pi}{6}-x\right)-\sqrt{3}cosx.sin4x\)

\(\Leftrightarrow sin4x\left(sinx+\sqrt{3}cosx\right)=\sqrt{2}sin\left(x+\frac{\pi}{3}\right)\)

\(\Leftrightarrow sin4x\left(\frac{1}{2}sinx+\frac{\sqrt{3}}{2}cosx\right)-\frac{\sqrt{2}}{2}sin\left(x+\frac{\pi}{3}\right)=0\)

\(\Leftrightarrow sin4x.sin\left(x+\frac{\pi}{3}\right)-\frac{\sqrt{2}}{2}sin\left(x+\frac{\pi}{3}\right)=0\)

\(\Leftrightarrow\left(sin4x-\frac{\sqrt{2}}{2}\right)sin\left(x+\frac{\pi}{3}\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sin4x=\frac{\sqrt{2}}{2}\\sin\left(x+\frac{\pi}{3}\right)=0\end{matrix}\right.\)

\(\Leftrightarrow...\)