Bài 3: Một số phương trình lượng giác thường gặp

Chọn C

\(\Rightarrow\sqrt{2}.sin\left(3x-\dfrac{\pi}{4}\right)-\sqrt{2}.sin\left(5x-\dfrac{\pi}{3}\right)=0\Leftrightarrow sin\left(3x-\dfrac{\pi}{4}\right)=sin\left(5x-\dfrac{\pi}{3}\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}3x-\dfrac{\pi}{4}+k2\pi=5x-\dfrac{\pi}{3}\\\pi-3x+\dfrac{\pi}{4}+k2\pi=5x-\dfrac{\pi}{3}\end{matrix}\right.\)

\(\left[{}\begin{matrix}x=\dfrac{\pi}{12}+k\pi\\x=\dfrac{19\pi}{96}+\dfrac{k\pi}{4}\end{matrix}\right.\); k\(\in Z\)

=>-cosx+căn3*sinx=căn 3

=>sin(x-pi/6)=căn 3/2

=>x-pi/6=pi/3+k2pi hoặc x-pi/6=2/3pi+k2pi

=>x=pi/2+k2pi hoặc x=5/6pi+k2pi

\(\Leftrightarrow4\left(sin2x+cos2x\right)=2\sqrt{2}\)

\(\Leftrightarrow sin2x+cos2x=\dfrac{\sqrt{2}}{2}\)

=>sin(2x+pi/4)=1/2

=>2x+pi/4=pi/6+k2pi hoặc 2x+pi/4=5/6pi+k2pi

=>x=-1/24pi+kpi hoặc x=7/24pi+kpi

\(\Leftrightarrow sin3x\cdot\dfrac{\sqrt{3}}{2}+\dfrac{1}{2}\cdot cos3x=1\)

=>\(sin\left(3x+\dfrac{pi}{6}\right)=1\)

=>3x+pi/6=pi/2+k2pi

=>3x=pi/3+k2pi

=>x=pi/9+k2pi/3

ĐKXĐ: x/2<>pi/2+kpi

=>x<>pi+k2pi

\(cos\left(x\right)+tan\left(\dfrac{x}{2}\right)=1\)

=>\(1-2\cdot sin^2\left(\dfrac{x}{2}\right)+\dfrac{sin\left(\dfrac{x}{2}\right)}{cos\left(\dfrac{x}{2}\right)}=1\)

\(\Leftrightarrow sin\left(\dfrac{x}{2}\right)\left(-2sin\left(\dfrac{x}{2}\right)+\dfrac{1}{cos\left(\dfrac{x}{2}\right)}\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{x}{2}=kpi\\-sinx+1=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=k2pi\\x=\dfrac{pi}{2}+k2pi\end{matrix}\right.\)

=>x-30 độ=30 độ+k*360 độ hoặc x-30 độ=150 độ+k*360 độ

=>x=60 độ+k*360 độ hoặc x=180 độ+k*360 độ