a) tan(2x - 15^o ) = 1 <=> 2x = 15^o + 45^o + k180^o

                                 <=> x = 30^o + k90^o ; k \(\in\)  Z

Do - 180^o < x < 90^o

      - 180^o < 30^o + k90^o < 90^o <=> - 2 < \(\frac{1}{3}\) + k < 1 <=> k \(\in\) { - 2 ; - 1 ; 0 }

Vậy các nghiệm của phương trình là z = - 150^o ; x = -60^o và x = 30^o .

b) cos3x = \(-\frac{1}{\sqrt{3}}\) <=> x = \(-\frac{\pi}{9}+k\frac{\pi}{3};k\in Z\)

Do \(-\frac{\pi}{2}< x< 0\) , ta có

\(-\frac{\pi}{2}< -\frac{\pi}{9}+k\frac{\pi}{3}< 0\) ↔ \(-\frac{7}{6}< k< \frac{1}{3}\) ↔ \(k\in\left\{-1;0\right\}\)

Vậy các nghiệm của phương trình là \(x=-\frac{4\pi}{9}\) và \(x=-\frac{\pi}{9}\)

@Nguyễn Việt Lâm giúp em với ạ

1.

a.

\(\Leftrightarrow sin\left(3x-30^0\right)=sin\left(45^0\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}3x-30^0=45^0+k360^0\\3x-30^0=135^0+k360^0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{75^0}{3}+k120^0\\x=\frac{165^0}{3}+k120^0\end{matrix}\right.\)

b.

\(sin\left(5x-\frac{\pi}{3}\right)=sin\left(2\pi-\frac{\pi}{4}-2x\right)\)

\(\Leftrightarrow sin\left(5x-\frac{\pi}{3}\right)=sin\left(-\frac{\pi}{4}-2x\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}5x-\frac{\pi}{3}=-\frac{\pi}{4}-2x+k2\pi\\5x-\frac{\pi}{3}=\frac{5\pi}{4}+2x+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{84}+\frac{k2\pi}{7}\\x=\frac{19\pi}{36}+\frac{k2\pi}{3}\end{matrix}\right.\)

c.

\(4x-\frac{\pi}{3}=k\pi\)

\(\Leftrightarrow x=\frac{\pi}{12}+\frac{k\pi}{4}\)

d.

\(sin\left(2x+\frac{\pi}{6}\right)=-1\)

\(\Leftrightarrow2x+\frac{\pi}{6}=-\frac{\pi}{2}+k2\pi\)

\(\Leftrightarrow x=-\frac{\pi}{3}+k\pi\)

Do \(x\in\left(-\frac{\pi}{4};2\pi\right)\Rightarrow-\frac{\pi}{4}< -\frac{\pi}{3}+k\pi< 2\pi\)

\(\Rightarrow\frac{1}{12}< k< \frac{7}{3}\Rightarrow k=\left\{1;2\right\}\)

\(\Rightarrow x=\left\{\frac{2\pi}{3};\frac{5\pi}{3}\right\}\)

e.

\(sin\left(x+\frac{\pi}{6}\right)=\frac{\sqrt{2}}{2}\Leftrightarrow\left[{}\begin{matrix}x+\frac{\pi}{6}=\frac{\pi}{4}+k2\pi\\x+\frac{\pi}{6}=\frac{3\pi}{4}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{12}+k2\pi\\x=\frac{7\pi}{12}+k2\pi\end{matrix}\right.\) \(\Rightarrow x=\left\{\frac{\pi}{12};\frac{7\pi}{12}\right\}\)

\(a)\;sinx = \frac{{\sqrt 3 }}{2}\)

Vì \(sin\frac{\pi }{3} = \frac{{\sqrt 3 }}{2}\) nên \(sinx = \frac{{\sqrt 3 }}{2} \Leftrightarrow sin\frac{\pi }{3} = sin\frac{\pi }{3}\) \( \Leftrightarrow \left[ \begin{array}{l}x = \frac{\pi }{3} + k2\pi ,k \in \mathbb{Z}\\x = \pi  - \frac{\pi }{3} + k2\pi ,k \in \mathbb{Z}\end{array} \right.\)

\( \Leftrightarrow \left[ \begin{array}{l}x = \frac{\pi }{3} + k2\pi ,k \in \mathbb{Z}\\x = \frac{{2\pi }}{3} + k2\pi ,k \in \mathbb{Z}\end{array} \right.\)

Vậy phương trình có nghiệm là \(x = \frac{\pi }{3} + k2\pi \) hoặc \(x = \frac{{2\pi }}{3} + k2\pi \)\(,k \in \mathbb{Z}\).

\(\begin{array}{l}b)\;sin(x + {30^o}) = sin(x + {60^o})\\ \Leftrightarrow \left[ \begin{array}{l}x + {30^o} = x + {60^o} + k{360^o},k \in \mathbb{Z}\\x + {30^o} = {180^o} - x - {60^o} + k{360^o},k \in \mathbb{Z}\end{array} \right.\\ \Leftrightarrow x = {45^o} + k{180^o},k \in \mathbb{Z}.\end{array}\)

Vậy phương trình có nghiệm là \(x = {45^o} + k{180^o},k \in \mathbb{Z}\).