\(\left\{{}\begin{matrix}x\in\left(0;\dfrac{\pi}{2}\right)\\sinx=\dfrac{\sqrt{3}}{2}\end{matrix}\right.\) \(\Rightarrow x=\dfrac{\pi}{3}\)
\(\Rightarrow\dfrac{x}{2}=\dfrac{\pi}{6}\Rightarrow cos\dfrac{x}{2}=cos\dfrac{\pi}{6}=\dfrac{\sqrt{3}}{2}\)
Cho x ∈ (0;\(\dfrac{\Pi}{2}\)) và sinx=\(\dfrac{\sqrt{3}}{2}\) . Khi đó cos \(\dfrac{x}{2}\) bằng
A.\(\dfrac{\sqrt{3}}{2}\)
B.\(\dfrac{1}{2}\)
C. \(-\dfrac{1}{2}\)
D.\(-\dfrac{\sqrt{3}}{2}\)
1) \(\dfrac{\sqrt{3}\cos x-2}{2\sin x-1}=0\)
2) \(\dfrac{\sqrt{3}\tan x-1}{2\cos x+\sqrt{3}}=0\)
Tính :
a) \(\sin\alpha,\) nếu \(\cos\alpha=-\dfrac{\sqrt{2}}{3}\) và \(\dfrac{\pi}{2}< \alpha< \pi\)
b) \(\cos\alpha\), nếu \(\tan\alpha=2\sqrt{2}\) và \(\pi< \alpha< \dfrac{3\pi}{2}\)
c) \(\tan\alpha\), nếu \(\sin\alpha=-\dfrac{2}{3}\) và \(\dfrac{3\pi}{2}< \alpha< 2\pi\)
d) \(\cot\alpha\), nếu \(\cos\alpha=-\dfrac{1}{4}\) và \(\dfrac{\pi}{2}< \alpha< \pi\)
Chứng minh các đẳng thức :
a) \(\tan3\alpha-\tan2\alpha-\tan\alpha=\tan\alpha\tan2\alpha\tan3\alpha\)
b) \(\dfrac{4\tan\alpha\left(1-\tan^2\alpha\right)}{\left(1+\tan^2\alpha\right)^2}=\sin4\alpha\)
c) \(\dfrac{1+\tan^4\alpha}{\tan^2\alpha+\cot^2\alpha}=\tan^2\alpha\)
d) \(\dfrac{\cos\alpha\sin\left(\alpha-3\right)-\sin\alpha\cos\left(\alpha-3\right)}{\cos\left(3-\dfrac{\pi}{6}\right)-\dfrac{1}{2}\sin3}=-\dfrac{2\tan3}{\sqrt{3}}\)
a, cho tan a=3 . tính gt của biểu thức
\(\dfrac{\sin a\cos a+\cos^2a}{2\sin^2a-\cos^2a}\)
b, c/m đẳng thức
\(\cot\left(\dfrac{\pi}{2}-x\right)\cos\left(\dfrac{\pi}{2}+x\right)+\dfrac{\sin\left(\pi-x\right)\cot x}{1-\sin^2x}=\cos x\)
Chứng minh các đồng nhất thức :
a) \(\dfrac{1-\cos x+\cos2x}{\sin2x-\sin x}=\cot x\)
b) \(\dfrac{\sin x+\sin\dfrac{x}{2}}{1+\cos x+\cos\dfrac{x}{2}}=\tan\dfrac{x}{2}\)
c) \(\dfrac{2\cos2x-\sin4x}{2\cos2x+\sin4x}=\tan^2\left(\dfrac{\pi}{4}-x\right)\)
d) \(\tan x-\tan y=\dfrac{\sin\left(x-y\right)}{\cos x\cos y}\)
Rút gọn các biểu thức :
a) \(\dfrac{\tan2\alpha}{\tan4\alpha-\tan2\alpha}\)
b) \(\sqrt{1+\sin\alpha}-\sqrt{1-\sin\alpha}\), với \(0< \alpha< \dfrac{\pi}{2}\)
c) \(\dfrac{3-4\cos2\alpha+\cos4\alpha}{3+4\cos2\alpha+\cos4\alpha}\)
d) \(\dfrac{\sin\alpha+\sin3\alpha+\sin5\alpha}{\cos\alpha+\cos3\alpha+\cos5\alpha}\)
Không sử dụng máy tính, hãy tính :
a) \(\cos\dfrac{22\pi}{3}\)
b) \(\sin\dfrac{23\pi}{4}\)
c) \(\sin\dfrac{25\pi}{3}-\tan\dfrac{10\pi}{3}\)
d) \(\cos^2\dfrac{\pi}{8}-\sin^2\dfrac{\pi}{8}\)
Cho \(\cos a=-\dfrac{\sqrt{5}}{3}\) với \(\pi< a< \dfrac{3\pi}{2}\)
Tính giá trị \(\tan\alpha\) ?
