giúp e vs
Bài tập 1: Giải các phương trình lượng giác sau:
a) \(\sin x = \frac{\sqrt{3}}{2}\)
b) \(2\cos x = -\sqrt{2}\)
c) \(\sqrt{3}\tan\left(\frac{x}{2} + 15^\circ\right) = 1\)
d) \(\cot(2x - 1) = \cot\frac{\pi}{5}\)
e) \(\sin 2x + \cos 4x = 0\)
f) \(\cos 3x = -\cos 7x\)
Bài tập 2: Giải các phương trình
a) \(\cos\left(2x + \frac{\pi}{6}\right) = 0\)
b) \(\cos\left(4x - \frac{\pi}{3}\right) = 1\)
c) \(\cos\left(\frac{\pi}{5} - x\right) = -1\)
d) \(\sin\left(3x + \frac{\pi}{3}\right) = 0\)
e) \(\sin\left(\frac{x}{2} - \frac{\pi}{4}\right) = 1\)
f) \(\sin\left(\frac{\pi}{6} + 2x\right) = -1\)
Bài 1:
a: \(\sin x=\frac{\sqrt3}{2}\)
=>\(\left[\begin{array}{l}x=\frac{\pi}{3}+k2\pi\\ x=\pi-\frac{\pi}{3}+k2\pi=\frac23\pi+k2\pi\end{array}\right.\)
b: \(2\cdot cosx=-\sqrt2\)
=>\(cosx=-\frac{\sqrt2}{2}\)
=>\(\left[\begin{array}{l}x=\frac34\pi+k2\pi\\ x=-\frac34\pi+k2\pi\end{array}\right.\)
c: \(\sqrt3\cdot\tan\left(\frac{x}{2}+15^0\right)=1\)
=>\(\tan\left(\frac12x+15^0\right)=\frac{1}{\sqrt3}\)
=>\(\frac12x+15^0=30^0+k\cdot180^0\)
=>\(\frac12x=15^0+k\cdot180^0\)
=>\(x=30^0+k\cdot360^0\)
d: \(\cot\left(2x-1\right)=\cot\left(\frac{\pi}{5}\right)\)
=>\(2x-1=\frac{\pi}{5}+k\pi\)
=>\(2x=\frac{\pi}{5}+1+k\pi\)
=>\(x=\frac{\pi}{10}+\frac12+\frac{k\pi}{2}\)
e: \(\sin2x+cos4x=0\)
=>\(cos4x=-\sin2x=\sin\left(-2x\right)\)
=>\(cos4x=cos\left(\frac{\pi}{2}+2x\right)\)
=>\(\left[\begin{array}{l}4x=2x+\frac{\pi}{2}+k2\pi\\ 4x=-2x-\frac{\pi}{2}+k2\pi\end{array}\right.\Rightarrow\left[\begin{array}{l}2x=\frac{\pi}{2}+k2\pi\\ 6x=-\frac{\pi}{2}+k2\pi\end{array}\right.\)
=>\(\left[\begin{array}{l}x=\frac{\pi}{4}+k\pi\\ x=-\frac{\pi}{12}+\frac{k\pi}{3}\end{array}\right.\)
f: cos3x=-cos7x
=>\(cos3x=cos\left(\pi-7x\right)\)
=>\(\left[\begin{array}{l}3x=\pi-7x+k2\pi\\ 3x=7x-\pi+k2\pi\end{array}\right.\Rightarrow\left[\begin{array}{l}10x=\pi+k2\pi\\ -4x=-\pi+k2\pi\end{array}\right.\)
=>\(\left[\begin{array}{l}x=\frac{\pi}{10}+\frac{k\pi}{5}\\ x=\frac14\pi-\frac{k\pi}{2}\end{array}\right.\)
Bài 2:
a: \(cos\left(2x+\frac{\pi}{6}\right)=0\)
=>\(2x+\frac{\pi}{6}=\frac{\pi}{2}+k\pi\)
=>\(2x=\frac{\pi}{2}+k\pi-\frac{\pi}{6}=\frac{\pi}{3}+k\pi\)
=>\(x=\frac{\pi}{6}+\frac{k\pi}{2}\)
b: \(cos\left(4x-\frac{\pi}{3}\right)=1\)
=>\(4x-\frac{\pi}{3}=k2\pi\)
=>\(4x=\frac{\pi}{3}+k2\pi\)
=>\(x=\frac{\pi}{12}+\frac{k\pi}{2}\)
c: \(cos\left(\frac{\pi}{5}-x\right)=-1\)
=>\(\frac{\pi}{5}-x=\pi+k2\pi\)
=>\(x=\frac{\pi}{5}-\pi-k2\pi=-\frac45\pi-k2\pi\)
d: \(\sin\left(3x+\frac{\pi}{3}\right)=0\)
=>\(3x+\frac{\pi}{3}=k\pi\)
=>\(3x=-\frac{\pi}{3}+k\pi\)
=>\(x=-\frac{\pi}{9}+\frac{k\pi}{3}\)
e: \(\sin\left(\frac{x}{2}-\frac{\pi}{4}\right)=1\)
=>\(\frac{x}{2}-\frac{\pi}{4}=\frac{\pi}{2}+k2\pi\)
=>\(\frac{x}{2}=\frac34\pi+k2\pi\)
=>\(x=\frac32\pi+k4\pi\)
f: \(\sin\left(2x+\frac{\pi}{6}\right)=-1\)
=>\(2x+\frac{\pi}{6}=-\frac{\pi}{2}+k2\pi\)
=>\(2x=-\frac{\pi}{2}-\frac{\pi}{6}+k2\pi=-\frac46\pi+k2\pi=-\frac23\pi+k2\pi\)
=>\(x=-\frac13\pi+k\pi\)
