Tính giá trị của biểu thức sau:
\(A=cos\dfrac{\pi}{7}cos\dfrac{4\pi}{7}cos\dfrac{5\pi}{7}\)
\(B=sin60^0.sin42^0.sin66^0.sin78^0\)
Tính giá trị của biểu thức sau:
\(A=cos\dfrac{\pi}{7}cos\dfrac{2\pi}{7}cos\dfrac{4\pi}{7}\)
\(A.sin\dfrac{\pi}{7}=sin\left(\dfrac{\pi}{7}\right)cos\left(\dfrac{\pi}{7}\right)cos\left(\dfrac{2\pi}{7}\right)cos\left(\dfrac{4\pi}{7}\right)\)
\(=\dfrac{1}{2}sin\left(\dfrac{2\pi}{7}\right)cos\left(\dfrac{2\pi}{7}\right)cos\left(\dfrac{4\pi}{7}\right)\)
\(=\dfrac{1}{4}sin\left(\dfrac{4\pi}{7}\right)cos\left(\dfrac{4\pi}{7}\right)\)
\(=\dfrac{1}{8}sin\left(\dfrac{8\pi}{7}\right)\)
\(=\dfrac{1}{8}sin\left(\pi+\dfrac{\pi}{7}\right)=\dfrac{1}{8}sin\left(-\dfrac{\pi}{7}\right)\)
\(=-\dfrac{1}{8}sin\left(\dfrac{\pi}{7}\right)\)
\(\Rightarrow A=-\dfrac{1}{8}\)
a) Biến đổi \(\sin\alpha-1\)thành tích
b) Rút gọn biểu thức \(P=\dfrac{\cos\alpha+2\cos3\alpha+\cos5a}{\sin\alpha+2\sin3\alpha+\sin5a}\)
c) Tính giá trị biểu thức \(P=\sin30.\cos60+\sin60.\cos30\)
d) Giá đúng của \(cos\dfrac{2\pi}{7}+\cos\dfrac{4\pi}{7}+\cos\dfrac{6\pi}{7}\)
e) Giá trị đúng của \(\tan\dfrac{\pi}{24}+\tan\dfrac{7\pi}{24}\)
a/\(sina-1=2sin\dfrac{a}{2}.cos\dfrac{a}{2}-sin^2\dfrac{a}{2}-cos^2\dfrac{a}{2}=-\left(sin\dfrac{a}{2}-cos\dfrac{a}{2}\right)^2\)
b/\(P=\dfrac{cosa+cos5a+2cos3a}{sina+sin5a+2sin3a}=\dfrac{2cos3a.cos2a+2cos3a}{2sin3a.cos2a+2sin3a}=\dfrac{2cos3a\left(cos2a+1\right)}{2sin3a\left(cos2a+1\right)}=cot3a\)
c/\(P=sin\left(30+60\right)=sin90=1\)
d/
\(A=cos\dfrac{2\pi}{7}+cos\dfrac{6\pi}{7}+cos\dfrac{4\pi}{7}\Rightarrow A.sin\dfrac{\pi}{7}=sin\dfrac{\pi}{7}.cos\dfrac{2\pi}{7}+sin\dfrac{\pi}{7}cos\dfrac{4\pi}{7}+sin\dfrac{\pi}{7}.cos\dfrac{6\pi}{7}\)
\(=\dfrac{1}{2}sin\dfrac{3\pi}{7}-\dfrac{1}{2}sin\dfrac{\pi}{7}+\dfrac{1}{2}sin\dfrac{5\pi}{7}-\dfrac{1}{2}sin\dfrac{3\pi}{7}+\dfrac{1}{2}sin\dfrac{7\pi}{7}-\dfrac{1}{2}sin\dfrac{5\pi}{7}\)
\(=-\dfrac{1}{2}sin\dfrac{\pi}{7}\Rightarrow A=-\dfrac{1}{2}\)
e/
\(tan\dfrac{\pi}{24}+tan\dfrac{7\pi}{24}=\dfrac{sin\dfrac{\pi}{24}}{cos\dfrac{\pi}{24}}+\dfrac{sin\dfrac{7\pi}{24}}{cos\dfrac{7\pi}{24}}=\dfrac{sin\dfrac{\pi}{24}cos\dfrac{7\pi}{24}+sin\dfrac{7\pi}{24}cos\dfrac{\pi}{24}}{cos\dfrac{\pi}{24}.cos\dfrac{7\pi}{24}}\)
\(=\dfrac{sin\left(\dfrac{\pi}{24}+\dfrac{7\pi}{24}\right)}{\dfrac{1}{2}cos\dfrac{\pi}{4}+\dfrac{1}{2}cos\dfrac{\pi}{3}}=\dfrac{2sin\dfrac{\pi}{3}}{cos\dfrac{\pi}{4}+cos\dfrac{\pi}{3}}=\dfrac{\sqrt{3}}{\dfrac{\sqrt{2}}{2}+\dfrac{1}{2}}=\dfrac{2\sqrt{3}}{\sqrt{2}+1}\)
sina - 1 = sina - sin\(\dfrac{\pi}{2}\)
Không dùng máy tính, tính giá trị của các biểu thức:
\(A = \cos {75^0}\cos {15^0}\);
\(B = \sin \frac{{5\pi }}{{12}}\cos \frac{{7\pi }}{{12}}\).
\(A = \cos {75^0}\cos {15^0} = \frac{1}{2}\left[ {\cos \left( {{{75}^0} - {{15}^0}} \right) + \cos \left( {{{75}^0} + {{15}^0}} \right)} \right] \\= \frac{1}{2}.\cos {60^0}.\cos {90^0} = 0\)
\(B = \sin \frac{{5\pi }}{{12}}\cos \frac{{7\pi }}{{12}} = \frac{1}{2}\left[ {\sin \left( {\frac{{5\pi }}{{12}} - \frac{{7\pi }}{{12}}} \right) + \sin \left( {\frac{{5\pi }}{{12}} + \frac{{7\pi }}{{12}}} \right)} \right] \\= \frac{1}{2}\sin \left( { - \frac{{2\pi }}{{12}}} \right).\sin \left( {\frac{{12\pi }}{{12}}} \right) = - \frac{1}{2}\sin \frac{\pi }{6}\sin \pi = 0\)
Tính giá trị của biểu thức sau : B= \(\dfrac{tan\left(\dfrac{21\pi}{2}-x\right).cos\left(38\pi-x\right).sin\left(x-7\pi\right)}{sin\left(\dfrac{13\pi}{2}-x\right).cos\left(x-2023\pi\right)}\)
Tính giá trị biểu thức \(cos\left(x+\dfrac{\Pi}{3}\right)\) biết \(sinx=\dfrac{1}{\sqrt{3}}\left(0< x< \dfrac{\Pi}{2}\right)\)
Tính :
a) \(4\left(\cos24^0+\cos48^0-\cos84^0-\cos12^0\right)\)
b) \(96\sqrt{3}\sin\dfrac{\pi}{48}\cos\dfrac{\pi}{48}\cos\dfrac{\pi}{24}\cos\dfrac{\pi}{12}\cos\dfrac{\pi}{6}\)
c) \(\tan9^0-\tan63^0+\tan81^0-\tan27^0\)
Giải các phương trình sau:
\(a,cos3x-4cos2x+3cosx-4=0\)
\(b,cos\left(x+\dfrac{\pi}{5}\right).cos\left(x-\dfrac{\pi}{5}\right)=cos\left(\dfrac{2\pi}{5}\right)\)
tính F=\(\sin^2\dfrac{\pi}{6}+\sin^2\dfrac{2\pi}{6}+...+\sin^2\dfrac{5\pi}{6}+\sin^2\pi\)
2/ biết \(\sin\beta=\dfrac{4}{5},0< \beta< \dfrac{\pi}{2}\) giá trị của biểu thúc a=\(\dfrac{\sqrt{3}\sin\left(\alpha+\beta\right)-\dfrac{4\cos\left(\alpha+\beta\right)}{\sqrt{3}}}{\sin\alpha}\)
Ta có \(F=sin^2\dfrac{\pi}{6}+...+sin^2\pi=\left(sin^2\dfrac{\pi}{6}+sin^2\dfrac{5\pi}{6}\right)+\left(sin^2\dfrac{2\pi}{6}+sin^2\dfrac{4\pi}{6}\right)+\left(sin^2\dfrac{3\pi}{6}+sin^2\pi\right)=\left(sin^2\dfrac{\pi}{6}+cos^2\dfrac{\pi}{6}\right)+\left(sin^2\dfrac{2\pi}{6}+cos^2\dfrac{2\pi}{6}\right)+\left(1+0\right)=1+1+1=3\)
Tính
A = \(\dfrac{1}{cos\dfrac{\pi}{7}}+\dfrac{1}{cos\dfrac{3\pi}{7}}+\dfrac{1}{cos\dfrac{5\pi}{7}}\)
A\(=\dfrac{cos\dfrac{5\pi}{7}.cos\dfrac{3\pi}{7}+cos\dfrac{5\pi}{7}.cos\dfrac{\pi}{7}+cos\dfrac{3\pi}{7}.cos\dfrac{\pi}{7}}{cos\dfrac{\pi}{7}.cos\dfrac{3\pi}{7}.cos\dfrac{5\pi}{7}}\)
Đặt tử là Y; mẫu là U
Có \(Y=\)\(cos\dfrac{5\pi}{7}.cos\dfrac{3\pi}{7}+\left(cos\dfrac{5\pi}{7}.cos\dfrac{\pi}{7}+cos\dfrac{3\pi}{7}.cos\dfrac{\pi}{7}\right)\)
\(=cos\left(\pi-\dfrac{2\pi}{7}\right).cos\left(\pi-\dfrac{4\pi}{7}\right)+cos\dfrac{\pi}{7}\left(cos\dfrac{5\pi}{7}+cos\dfrac{3\pi}{7}\right)\)
\(=cos\dfrac{2\pi}{7}.cos\dfrac{4\pi}{7}+cos\dfrac{\pi}{7}.2cos\dfrac{4\pi}{7}.cos\dfrac{\pi}{7}\)\(=cos\dfrac{2\pi}{7}.cos\dfrac{4\pi}{7}+2.cos^2\dfrac{\pi}{7}.cos\dfrac{4\pi}{7}\)
\(=cos\dfrac{2\pi}{7}.cos\dfrac{4\pi}{7}+\left(cos\dfrac{2\pi}{7}+1\right).cos\dfrac{4\pi}{7}\)\(=2.cos\dfrac{2\pi}{7}.cos\dfrac{4\pi}{7}+cos\dfrac{4\pi}{7}\)
\(=cos\dfrac{6\pi}{7}+cos\dfrac{2\pi}{7}+cos\dfrac{4\pi}{7}\)
\(\Rightarrow sin\dfrac{\pi}{7}.Y=sin\dfrac{\pi}{7}.cos\dfrac{2\pi}{7}+sin\dfrac{\pi}{7}.cos\dfrac{4\pi}{7}+sin\dfrac{\pi}{7}.cos\dfrac{6\pi}{7}\)
\(=\dfrac{1}{2}\left(-sin\dfrac{\pi}{7}+sin\dfrac{3\pi}{7}\right)+\dfrac{1}{2}\left(-sin\dfrac{3\pi}{7}+sin\dfrac{5\pi}{7}\right)+\dfrac{1}{2}\left(-sin\dfrac{5\pi}{7}+sin\pi\right)\)
\(=\dfrac{1}{2}\left(sin\pi-sin\dfrac{\pi}{7}\right)\)\(=-\dfrac{1}{2}sin\dfrac{\pi}{7}\)
\(\Rightarrow Y=-\dfrac{1}{2}\)
Có \(sin\dfrac{\pi}{7}.U=sin\dfrac{\pi}{7}.cos\dfrac{\pi}{7}.cos\dfrac{3\pi}{5}.cos\dfrac{5\pi}{7}\)
\(=\dfrac{1}{2}.sin\dfrac{2\pi}{7}.cos\left(\pi-\dfrac{2\pi}{7}\right).cos\dfrac{3\pi}{5}\)
\(=-\dfrac{1}{4}.sin\dfrac{4\pi}{7}.cos\left(\pi-\dfrac{4\pi}{5}\right)\)
\(=\dfrac{1}{8}.sin\dfrac{8\pi}{7}\)\(=\dfrac{1}{8}.sin\left(\pi+\dfrac{\pi}{7}\right)=-\dfrac{1}{8}.sin\dfrac{\pi}{7}\)
\(\Rightarrow U=-\dfrac{1}{8}\)
Vậy \(A=\dfrac{Y}{U}=4\)