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Bài 4. Giải các phương trình:
a) \(\sin (x + 1) = {2 \over 3}\)
b) \({\sin ^2}2x = {1 \over 2}\)
c) \({\cot ^2}{x \over 2} = {1 \over 3}\)
d) \(\tan ({\pi \over {12}} + 12x) = – \sqrt 3 \)
a) Ta có:
\(\eqalign{
& \sin (x + 1) = {2 \over 3} \cr
& \Leftrightarrow \left[ \matrix{
x + 1 = \arcsin {2 \over 3} + k2\pi \hfill \cr
x + 1 = \pi – \arcsin {2 \over 3} + k2\pi \hfill \cr} \right. \cr
& \Leftrightarrow \left[ \matrix{
x = – 1 + \arcsin {2 \over 3} + k2\pi \hfill \cr
x = – 1 + \pi – \arcsin {2 \over 3} + k2\pi \hfill \cr} \right.;k \in \mathbb{Z} \cr} \)
b) Ta có:
\(\eqalign{
& {\sin ^2}2x = {1 \over 2} \Leftrightarrow {{1 – \cos 4x} \over 2} = {1 \over 2} \cr
& \Leftrightarrow \cos 4x = 0 \Leftrightarrow 4x = {\pi \over 2} + k\pi \cr
& \Leftrightarrow x = {\pi \over 8} + k{\pi \over 4},k \in \mathbb{Z} \cr} \)
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c) Ta có:
\(\eqalign{
& {\cot ^2}{x \over 2} = {1 \over 3} \Leftrightarrow \left[ \matrix{
\cot {x \over 2} = {{\sqrt 3 } \over 3}(1) \hfill \cr
\cot {x \over 2} = – {{\sqrt 3 } \over 3}(2) \hfill \cr} \right. \cr
& (1) \Leftrightarrow \cot {x \over 2} = \cot {\pi \over 3} \Leftrightarrow {x \over 2} = {\pi \over 3} + k\pi \cr
& \Leftrightarrow x = {{2\pi } \over 3} + k2\pi ,k \in \mathbb{Z} \cr
& (2) \Leftrightarrow \cot {x \over 2} = \cot ( – {\pi \over 3}) \Leftrightarrow {x \over 2} = – {\pi \over 3} + k\pi \cr
& \Leftrightarrow x = – {{2\pi } \over 3} + k2\pi ;k \in \mathbb{Z} \cr} \)
d) Ta có:
\( \tan ({\pi \over {12}} + 12x) = – \sqrt 3\)
\(\Leftrightarrow \tan ({\pi \over {12}} + 12x ) = \tan ({{ – \pi } \over 3})\)
\(\Leftrightarrow {\pi \over {12}} + 12x = {{ – \pi } \over 3} + k\pi\)
\(\Leftrightarrow x = – {{5\pi } \over {144}} + k{\pi \over {12}},k \in \mathbb{Z} \)
Vậy nghiệm của phương trình đã cho là: \(x = {{ – 5\pi } \over {144}} + {{k\pi } \over {12}},k \in \mathbb{Z}\)