Bài 1 (Trang 28 SGK Toán Đại số & Giải tích 11):

Giải các phương trình sau:

a) $\sin \left(x+2\right) = \frac{1}{3};$

b) sin 3x = 1

c) $\sin \left(\frac{3x}{3} - \frac{\pi }{3}\right) = 0;$

d) $\sin \left(2x +\hspace{0.17em}20°\right) = \frac{-\sqrt{3}}{2}.$

Giải:

a) Ta có sin (x + 2) = $\frac{1}{3}$$$\begin{gathered} \iff {[}_{x + 2 = \mathrm{\pi } - \arcsin \frac{1}{3} + \mathrm{k}2\mathrm{\pi }}^{x + 2 = arc\sin \frac{1}{3} + k2\mathrm{\pi }} (k\in \mathrm{\mathbb{Z}}) \\ {\iff }_{x = \mathrm{\pi } - \arcsin \frac{1}{3} -2 + \mathrm{k}2\mathrm{\pi }}^{x = arc\sin \frac{1}{3} - 2 + k2\mathrm{\pi }} (k\in \mathrm{\mathbb{Z}}) \end{gathered}$$

b) sin3x = 1 $\iff 3x = \frac{\mathrm{\pi }}{2} + k2\mathrm{\pi } \iff \mathrm{x} = \frac{\mathrm{\pi }}{6} + \mathrm{k}\frac{2\mathrm{\pi }}{3}; \mathrm{k} \in \mathrm{\mathbb{Z}}$

c) $\sin \left(\frac{2x}{3} -\frac{\mathrm{\pi }}{3}\right) = 0 \iff \frac{2x}{3}- \frac{\mathrm{\pi }}{3}= k\mathrm{\pi } \iff \mathrm{x} = \frac{\mathrm{\pi }}{2}+ \mathrm{k}\frac{3\mathrm{\pi }}{2}; \mathrm{k} \in \mathrm{\mathbb{Z}}$

d) sin$(2x + {20}^0) = -\frac{\sqrt{3}}{2} \Rightarrow \sin (2x + {20}^0) = \sin (-{60}^0)$

$\iff {[}_{2x + {20}^0 = {180}^0 - (-{60}^0) + k{360}^0}^{2x + {20}^0 = -{60}^0 + k{360}^0}\iff {[}_{x = {110}^0 + k{180}^0}^{x = -{40}^0 + k{180}^0}$

1. Giải các phương trình sau:

a) sin (x+2) =$\frac{1}{3}$;                       b) sin3x = 1

c) $\sin \left(\frac{2x}{3} - \frac{\mathrm{\pi }}{3}\right) = 0$             d) $\sin (2x +{20}^0) = \frac{-\sqrt{3}}{2}$.

Giải:

a) Ta có sin (x + 2) = $\frac{1}{3}$$$\begin{gathered} \iff {[}_{x + 2 = \mathrm{\pi } - \arcsin \frac{1}{3} + \mathrm{k}2\mathrm{\pi }}^{x + 2 = arc\sin \frac{1}{3} + k2\mathrm{\pi }} (k\in \mathrm{\mathbb{Z}}) \\ {\iff }_{x = \mathrm{\pi } - \arcsin \frac{1}{3} -2 + \mathrm{k}2\mathrm{\pi }}^{x = arc\sin \frac{1}{3} - 2 + k2\mathrm{\pi }} (k\in \mathrm{\mathbb{Z}}) \end{gathered}$$

b) sin3x = 1 $\iff 3x = \frac{\mathrm{\pi }}{2} + k2\mathrm{\pi } \iff \mathrm{x} = \frac{\mathrm{\pi }}{6} + \mathrm{k}\frac{2\mathrm{\pi }}{3}; \mathrm{k} \in \mathrm{\mathbb{Z}}$

c) $\sin \left(\frac{2x}{3} -\frac{\mathrm{\pi }}{3}\right) = 0 \iff \frac{2x}{3}- \frac{\mathrm{\pi }}{3}= k\mathrm{\pi } \iff \mathrm{x} = \frac{\mathrm{\pi }}{2}+ \mathrm{k}\frac{3\mathrm{\pi }}{2}; \mathrm{k} \in \mathrm{\mathbb{Z}}$

d) sin$(2x + {20}^0) = -\frac{\sqrt{3}}{2} \Rightarrow \sin (2x + {20}^0) = \sin (-{60}^0)$

$\iff {[}_{2x + {20}^0 = {180}^0 - (-{60}^0) + k{360}^0}^{2x + {20}^0 = -{60}^0 + k{360}^0}\iff {[}_{x = {110}^0 + k{180}^0}^{x = -{40}^0 + k{180}^0}$