Bài tập tự luận phương trình lượng giác

Bài 1: Giải phương trì̀h : \(\cos 2 x+3 \sin x-2=0\)

\(\begin{array}{c}\Leftrightarrow 1-2 \sin ^{2} x+3 \sin x-2=0 \Leftrightarrow 2 \sin ^{2} x-3 \sin x+1=0 \\ \Leftrightarrow\left[\begin{array}{l}\sin x=1 \\ \sin x=\frac{1}{2}\end{array} \Leftrightarrow\left[\begin{array}{l}x=\frac{\pi}{2}+k 2 \pi \\ x=\frac{\pi}{6}+k 2 \pi, k \in \mathbb{Z} \\ x=\frac{5 \pi}{6}+k 2 \pi\end{array}\right.\right.\end{array}\)

Bài 2: Giải phương trình: \(\sqrt{3} \sin x+\cos x=\sqrt{2}\)

\(\begin{array}{l}\sqrt{3} \sin x+\cos x=\sqrt{2} \Leftrightarrow \frac{\sqrt{3}}{2} \sin x+\frac{1}{2} \cos x=\frac{\sqrt{2}}{2} \\ \Leftrightarrow \sin x \cos \frac{\pi}{6}+\cos x \sin \frac{\pi}{6}=\frac{\sqrt{2}}{2} \Leftrightarrow \sin \left(x+\frac{\pi}{6}\right)=\sin \frac{\pi}{4} \\ \Leftrightarrow\left[\begin{array}{l}x+\frac{\pi}{6}=\frac{\pi}{4}+k 2 \pi \\ x+\frac{\pi}{6}=\frac{3 \pi}{4}+k 2 \pi\end{array} \Leftrightarrow\left[\begin{array}{l}x=\frac{\pi}{12}+k 2 \pi \\ x=\frac{7 \pi}{12}+k 2 \pi\end{array}, k \in \mathbb{Z}\right.\right.\end{array}\)

Bài 3: Giải phương trình : \(\sqrt{3} \sin x-\cos x=\sqrt{2}\)

Bài 4: Giải phương trình : \(2 \sin ^{2} x+3 \sin x \cos x-5 \cos ^{2} x=0\)

\(\begin{array}{l}\Leftrightarrow 2 \tan ^{2} x+3 \tan x-5=0 \\ \Leftrightarrow\left[\begin{array}{c}\tan x=1 \\ \tan x=-\frac{5}{2}\end{array} \Leftrightarrow\left[\begin{array}{c}x=\frac{\pi}{4}+k \pi \\ x=\arctan \left(-\frac{5}{2}\right)+k \pi\end{array}, k \in \mathbb{Z}\right.\right. \\\end{array}\)

Bài 5: Giải phương trình : \(3(\sin 5 x-\cos x)=4(\sin x+\cos 5 x)\)

\(\begin{array}{l}\Leftrightarrow 3 \sin 5 x-4 \cos 5 x=4 \sin x+3 \cos x \\ \Leftrightarrow \frac{3}{5} \sin 5 x-\frac{4}{5} \cos 5 x=\frac{4}{5} \sin x+\frac{3}{5} \cos x \\ \Leftrightarrow \sin 5 x \cos \alpha-\cos 5 x \sin \alpha=\sin x \sin \alpha+\cos x \cos \alpha,\left(\frac{3}{5}=\cos \alpha, \frac{4}{5}=\sin \alpha\right) \\ \Leftrightarrow \sin (5 x-\alpha)=\cos (x-\alpha) \Leftrightarrow \sin (5 x-\alpha)=\sin \left(\frac{\pi}{2}-x+\alpha\right) \\ \Leftrightarrow\left[\begin{array}{c}5 x-\alpha=\frac{\pi}{2}-x+\alpha+k 2 \pi \\ 5 x-\alpha=\pi-\frac{\pi}{2}+x-\alpha+k 2 \pi\end{array} \Leftrightarrow\left[\begin{array}{c}x=\frac{\pi}{12}+\frac{\alpha}{3}+k \frac{\pi}{3} \\ x=\frac{\pi}{8}+k \frac{\pi}{2}\end{array}\right.\right.\end{array}\)

Bài 6: Giải phương trình : \(3 \sin 3 x-\sqrt{3} \cos 9 x=1+4 \sin ^{3} 3 x\)

\(\begin{array}{l}\Leftrightarrow\left(3 \sin 3 x-4 \sin ^{3} 3 x\right)-\sqrt{3} \cos 9 x=1 \\ \Leftrightarrow \sin 9 x-\sqrt{3} \cos 9 x=1 \Leftrightarrow \sin \left(9 x-\frac{\pi}{3}\right)=\sin \frac{\pi}{6} \Leftrightarrow\left[\begin{array}{l}x=\frac{\pi}{18}+k \frac{2 \pi}{9} \\ x=\frac{7 \pi}{54}+k \frac{2 \pi}{9}\end{array}\right.\end{array}\)

Bài 7: Giải phương trình : \(\tan x-\sin 2 x-\cos 2 x+2\left(2 \cos x-\frac{1}{\cos x}\right)=0\)

Điều kiện: \(\cos x \neq 0 \Leftrightarrow x \neq \frac{\pi}{2}+k \pi\)

Bài 8: Giải phương trình : \(8 \sin x=\frac{\sqrt{3}}{\cos x}+\frac{1}{\sin x}\)

Điều kiện: \(\sin 2 x \neq 0 \Leftrightarrow x \neq k \frac{\pi}{2}\)

\(\begin{array}{l}\Leftrightarrow \cos 3 x=\frac{1}{2} \cos x-\frac{\sqrt{3}}{2} \sin x \Leftrightarrow \cos 3 x=\cos \left(x+\frac{\pi}{3}\right) \Leftrightarrow\left[\begin{array}{c}x=\frac{\pi}{6}+k \pi \\ x=-\frac{\pi}{12}+k \frac{\pi}{2}\end{array}\right. \\ \text { C2 }\left({ }^{*}\right) \Leftrightarrow 8 \sin ^{2} x \cos x=\sqrt{3} \sin x+\cos x \Leftrightarrow 8\left(1-\cos ^{2} x\right) \cos x=\sqrt{3} \sin x+\cos x \\ \Leftrightarrow 8 \cos x-8 \cos ^{3} x=\sqrt{3} \sin x-3 \cos x \Leftrightarrow 6 \cos x-8 \cos ^{3} x=\sqrt{3} \sin x-\cos x \\ \Leftrightarrow 4 \cos ^{3} x-3 \cos x=\frac{1}{2} \cos x-\frac{\sqrt{3}}{2} \sin x \Leftrightarrow \cos 3 x=\cos \left(x+\frac{\pi}{3}\right) \\ \Leftrightarrow\left[\begin{array}{c}x=\frac{\pi}{6}+k \pi \\ x=-\frac{\pi}{12}+k \frac{\pi}{2}\end{array}\right. \\\end{array}\)

Bài 9: Giải phương trình : \(9 \sin x+6 \cos x-3 \sin 2 x+\cos 2 x=8\)

\(\begin{array}{l} \Leftrightarrow 6 \sin x \cos x-6 \cos x+2 \sin ^{2} x-9 \sin x+7=0 \\ \Leftrightarrow 6 \cos x(\sin x-1)+(\sin x-1)(2 \sin x-7)=0 \\ \quad \Leftrightarrow(\sin x-1)(6 \cos x+2 \sin x-7)=0 \\ \Leftrightarrow\left[\begin{array}{c}\sin x=1 \\ 6 \cos x+2 \sin x=7\end{array} \Leftrightarrow x=\frac{\pi}{2}+k 2 \pi\right.\end{array}\)

Bài 10: Giải phương trình : \(\sin 2 x+2 \cos 2 x=1+\sin x-4 \cos x\)

\(\begin{array}{c}\Leftrightarrow 2 \sin x \cos x+2\left(2 \cos ^{2} x-1\right)-1-\sin x+4 \cos x=0 \\ \Leftrightarrow \sin x(2 \cos x-1)+4 \cos ^{2} x+4 \cos x-3=0\end{array}\)