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\(2sin\left(\frac{\pi}{2}+x\right)+sin\left(3\pi-x\right)+sin\left(\frac{3\pi}{2}+x\right)+cos\left(\frac{\pi}{2}+x\right)\)
\(=2cosx+sinx-cosx-sinx\)
\(=cosx\)
\(A=cosa\left(sinb.cosc-cosb.sinc\right)+cosb\left(sinc.cosa-cosc.sina\right)+cosc\left(sinacosb-cosasinb\right)\)
\(A=cosasinbcosc-cosacosbsinc+cosacosbsinc-sinacosbcosc+sinacosbcosc-cosasinbcosc\)
\(A=0\)
\(B=sin^2x+\frac{1}{2}\left(cos\frac{2\pi}{3}+cos2x\right)\)
\(B=\frac{1}{2}-\frac{1}{2}cos2x-\frac{1}{4}+\frac{1}{2}cos2x\)
\(B=\frac{1}{4}\)
\(C=\frac{1}{2}-\frac{1}{2}cos2x+\frac{1}{2}-\frac{1}{2}cos\left(\frac{4\pi}{3}+2x\right)+\frac{1}{2}-\frac{1}{2}cos\left(\frac{4\pi}{3}-2x\right)\)
\(C=\frac{3}{2}-\frac{1}{2}cos2x-\frac{1}{2}\left(cos\left(\frac{4\pi}{3}+2x\right)+cos\left(\frac{4\pi}{3}-2x\right)\right)\)
\(C=\frac{3}{2}-\frac{1}{2}cos2x-cos\frac{4\pi}{3}.cos2x\)
\(C=\frac{3}{2}-\frac{1}{2}cos2x+\frac{1}{2}cos2x\)
\(C=\frac{3}{2}\)
\(D=\frac{1}{2}\left[\sqrt{2}sin\left(\frac{\pi}{4}+x\right)\right]^2-sin^2x-sinx.\sqrt{2}cos\left(\frac{\pi}{4}+x\right)\)
\(D=\frac{1}{2}\left(sinx+cosx\right)^2-sin^2x-sinx\left(sinx-cosx\right)\)
\(D=\frac{1}{2}\left(1+2sinx.cosx\right)-sin^2x-sin^2x+sinx.cosx\)
\(D=\frac{1}{2}+sinxcosx+sinxcosx=\frac{1}{2}+sin2x\)
Góc độ cao của thang dựa vào tường là 60º và chân thang cách tường 4,6 m. Chiều dài của thang là
\(=sin\left(x+\frac{\pi}{2}+42\pi\right)+cos\left(206\pi+\pi+x\right)+sin^2\left(32\pi+\pi+x\right)+sin^2\left(x+\frac{\pi}{2}-2\pi\right)\)
\(=sin\left(x+\frac{\pi}{2}\right)+cos\left(\pi+x\right)+sin^2\left(\pi+x\right)+sin^2\left(x+\frac{\pi}{2}\right)\)
\(=cosx-cosx+sin^2x+cos^2x\)
\(=sin^2x+cos^2x=1\)
\(A=2cosx-3cosx-sin\left(3\pi+\frac{\pi}{2}-x\right)+tan\left(\pi+\frac{\pi}{2}-x\right)\)
\(A=-cosx+sin\left(\frac{\pi}{2}-x\right)+tan\left(\frac{\pi}{2}-x\right)\)
\(A=-cosx+cosx+cotx=cotx\)
\(B=2cosx+sin\left(4\pi+\pi-x\right)+sin\left(2\pi-\frac{\pi}{2}+x\right)-sinx\)
\(B=2cosx+sin\left(\pi-x\right)+sin\left(-\frac{\pi}{2}+x\right)-sinx\)
\(B=2cosx+sinx-sin\left(\frac{\pi}{2}-x\right)-sinx\)
\(B=2cosx-cosx=cosx\)
a/ \(\pi< x< \frac{3\pi}{2}\Rightarrow sinx< 0\)
\(\Rightarrow sinx=-\sqrt{1-cos^2x}=-\frac{5}{13}\)
\(sin\left(\frac{\pi}{3}-x\right)=sin\frac{\pi}{3}cosx-cos\frac{\pi}{3}sinx=\frac{\sqrt{3}}{2}.\left(-\frac{12}{13}\right)-\frac{1}{2}.\left(-\frac{5}{13}\right)=\frac{5-12\sqrt{3}}{26}\)
b/ \(\pi< x< \frac{3\pi}{2}\Rightarrow cosx< 0\)
\(\Rightarrow cosx=-\sqrt{1-sin^2x}=-\frac{3}{5}\)
\(cot\left(x-\frac{\pi}{4}\right)=\frac{cos\left(x-\frac{\pi}{4}\right)}{sin\left(x-\frac{\pi}{4}\right)}=\frac{sinx+cosx}{sinx-cosx}=7\)
c/ \(cot\left(\frac{5\pi}{2}-x\right)=cot\left(2\pi+\frac{\pi}{2}-x\right)=tanx=2\)
\(\Rightarrow tan\left(x+\frac{\pi}{4}\right)=\frac{tanx+tan\frac{\pi}{4}}{1-tanx.tan\frac{\pi}{4}}=\frac{2+1}{1-2.1}=-3\)
\(P=sin^4x+\left(sin^2\left(x+\frac{\pi}{4}\right)\right)^2+cos^4x+\left(cos^2\left(x+\frac{\pi}{4}\right)\right)^2\)
\(=\left(\frac{1}{2}-\frac{1}{2}cos2x\right)^2+\left(\frac{1}{2}-\frac{1}{2}cos\left(2x+\frac{\pi}{2}\right)\right)^2+\left(\frac{1}{2}+\frac{1}{2}cos2x\right)^2+\left(\frac{1}{2}+\frac{1}{2}cos\left(2x+\frac{\pi}{4}\right)\right)^2\)
\(=\frac{1}{4}-\frac{1}{2}cos2x+\frac{1}{4}cos^22x+\frac{1}{4}+\frac{1}{2}sin2x+\frac{1}{4}sin^22x+\frac{1}{4}+\frac{1}{2}cos2x+\frac{1}{4}cos^22x+\frac{1}{4}-\frac{1}{2}sin2x+\frac{1}{4}sin^22x\)
\(=1+\frac{1}{2}\left(sin^22x+cos^22x\right)=\frac{3}{2}\)
\(cos\left(2x+\frac{\pi}{6}\right)cos\left(2x-\frac{\pi}{6}\right)=\frac{1}{2}\left(cos4x+cos\frac{\pi}{3}\right)=\frac{1}{2}\left(cos4x+\frac{1}{2}\right)\)
\(sin\left(x+\frac{\pi}{6}\right)sin\left(x-\frac{\pi}{6}\right)=\frac{1}{2}\left(cos\frac{\pi}{3}-cos2x\right)=\frac{1}{2}\left(\frac{1}{2}-cos2x\right)\)
\(\Rightarrow C=\frac{1}{2}sinx.cos4x+\frac{1}{4}sinx+\frac{1}{4}sin3x-\frac{1}{2}sin3x.cos2x\)
\(=\frac{1}{4}sin5x-\frac{1}{4}sin3x+\frac{1}{4}sinx+\frac{1}{4}sin3x-\frac{1}{4}sin5x+\frac{1}{4}sinx\)
\(=\frac{1}{2}sinx\)
\(=cos\left(4\pi+\pi+x\right)+sin\left(4\pi+\frac{\pi}{2}-x\right)-tan\left(\pi+\frac{\pi}{2}+x\right).cot\left(\pi+\frac{\pi}{2}-x\right)\)
\(=cos\left(\pi+x\right)+sin\left(\frac{\pi}{2}-x\right)-tan\left(\frac{\pi}{2}+x\right).cot\left(\frac{\pi}{2}-x\right)\)
\(=-cosx+cosx-\left(-cotx\right).tanx\)
\(=1\)
\(sin\left(x+\frac{\pi}{3}\right)-sin\left(x-\frac{\pi}{3}\right)=sin\left(6\pi+\frac{\pi}{2}\right)\)
\(\Leftrightarrow2cosx.sin\frac{\pi}{3}=sin\left(\frac{\pi}{2}\right)\)
\(\Leftrightarrow2cosx.\frac{\sqrt{3}}{2}=1\)
\(\Rightarrow cosx=\frac{1}{\sqrt{3}}\)