c.

\(\Leftrightarrow cos\left(x+12^0\right)+cos\left(90^0-78^0+x\right)=1\)

\(\Leftrightarrow2cos\left(x+12^0\right)=1\)

\(\Leftrightarrow cos\left(x+12^0\right)=\dfrac{1}{2}\)

\(\Leftrightarrow\left[{}\begin{matrix}x+12^0=60^0+k360^0\\x+12^0=-60^0+k360^0\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=48^0+k360^0\\x=-72^0+k360^0\end{matrix}\right.\)

2.

Do \(-1\le sin\left(3x-27^0\right)\le1\) nên pt có nghiệm khi:

\(\left\{{}\begin{matrix}2m^2+m\ge-1\\2m^2+m\le1\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}2m^2+m+1\ge0\left(luôn-đúng\right)\\2m^2+m-1\le0\end{matrix}\right.\)

\(\Rightarrow-1\le m\le\dfrac{1}{2}\)

a.

\(\Rightarrow\left[{}\begin{matrix}x+15^0=arccos\left(\dfrac{2}{5}\right)+k360^0\\x+15^0=-arccos\left(\dfrac{2}{5}\right)+k360^0\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=-15^0+arccos\left(\dfrac{2}{5}\right)+k360^0\\x=-15^0-arccos\left(\dfrac{2}{5}\right)+k360^0\end{matrix}\right.\)

b.

\(2x-10^0=arccot\left(4\right)+k180^0\)

\(\Rightarrow x=5^0+\dfrac{1}{2}arccot\left(4\right)+k90^0\)

2.

Phương trình \(sin\left(3x-27^o\right)=2m^2+m\) có nghiệm khi:

\(2m^2+m\in\left[-1;1\right]\)

\(\Leftrightarrow\left\{{}\begin{matrix}2m^2+m\le1\\2m^2+m\ge-1\end{matrix}\right.\)

\(\Leftrightarrow\left(m+1\right)\left(2m-1\right)\le0\)

\(\Leftrightarrow-1\le m\le\dfrac{1}{2}\)

Bạn xem lại đề bài

a) \(2\cos x =  - \sqrt 2  \Leftrightarrow \cos x =  - \frac{{\sqrt 2 }}{2}\;\; \Leftrightarrow \cos x = \cos \frac{\pi }{4} \Leftrightarrow \;\left[ {\begin{array}{*{20}{c}}{x = \frac{\pi }{4} + k2\pi }\\{x = \pi  - \frac{\pi }{4} + k2\pi }\end{array}} \right.\;\;\;\;\;\; \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{x = \frac{\pi }{4} + k2\pi }\\{x = \frac{{3\pi }}{4} + k2\pi }\end{array}\;\left( {k \in \mathbb{Z}} \right)} \right.\)

b) \(\cos 3x - \sin 5x = 0\;\;\;\; \Leftrightarrow \cos 3x = \sin 5x\;\;\;\; \Leftrightarrow \cos 3x = \cos \left( {\frac{\pi }{2} - 5x} \right)\;\;\)

\( \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{3x = \frac{\pi }{2} - 5x + k2\pi }\\{3x =  - \frac{\pi }{2} + 5x + k2\pi }\end{array}} \right.\;\;\;\;\; \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{8x = \frac{\pi }{2} + k2\pi }\\{ - 2x =  - \frac{\pi }{2} + k2\pi }\end{array}} \right.\;\; \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{x = \frac{\pi }{{16}} + \frac{{k\pi }}{4}}\\{x = \frac{\pi }{4} - k\pi }\end{array}} \right.\;\;\left( {k \in \mathbb{Z}} \right)\)

a) Vì \(\sin \frac{\pi }{6} = \frac{1}{2}\) nên ta có phương trình \(sin2x = \sin \frac{\pi }{6}\)

\( \Leftrightarrow \left[ \begin{array}{l}2x = \frac{\pi }{6} + k2\pi \\2x = \pi  - \frac{\pi }{6} + k2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = \frac{\pi }{{12}} + k\pi \\x = \frac{{5\pi }}{{12}} + k\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right)\)

\(\begin{array}{l}b,\,\,sin(x - \frac{\pi }{7}) = sin\frac{{2\pi }}{7}\\ \Leftrightarrow \left[ \begin{array}{l}x - \frac{\pi }{7} = \frac{{2\pi }}{7} + k2\pi \\x - \frac{\pi }{7} = \pi  - \frac{{2\pi }}{7} + k2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = \frac{{3\pi }}{7} + k2\pi \\x = \frac{{6\pi }}{7} + k2\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right)\end{array}\)

\(\begin{array}{l}\;c)\;sin4x - cos\left( {x + \frac{\pi }{6}} \right) = 0\\ \Leftrightarrow sin4x = cos\left( {x + \frac{\pi }{6}} \right)\\ \Leftrightarrow sin4x = \sin \left( {\frac{\pi }{2} - x - \frac{\pi }{6}} \right)\\ \Leftrightarrow sin4x = \sin \left( {\frac{\pi }{3} - x} \right)\\ \Leftrightarrow \left[ \begin{array}{l}4x = \frac{\pi }{3} - x + k2\pi \\4x = \pi  - \frac{\pi }{3} + x + k2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = \frac{\pi }{{15}} + k\frac{{2\pi }}{5}\\x = \frac{{2\pi }}{9} + k\frac{{2\pi }}{3}\end{array} \right.\left( {k \in \mathbb{Z}} \right)\end{array}\)

tham khảo:

(sinx + sin5x) + (sin2x + sin4x) + 4sin3x = 0

⇔ 2sin3x . cos2x + 2sin3x . cosx + 4sin3x = 0

⇔ 2sin3x (cos2x + cosx + 2sin3x) = 0

⇔ \(\left[{}\begin{matrix}sin3x=0\left(1\right)\\cos2x+cosx+2sin3x=0\left(2\right)\end{matrix}\right.\)

(1) ⇔ ...

(2) ⇔ \(2cos\dfrac{3x}{2}.cos\dfrac{x}{2}+4sin\dfrac{3x}{2}.cos\dfrac{3x}{2}=0\)

⇔ \(\left[{}\begin{matrix}cos\dfrac{3x}{2}=0\left(\alpha\right)\\cos\dfrac{x}{2}+2sin\dfrac{3x}{2}=0\left(\beta\right)\end{matrix}\right.\)

Giải \(\left(\alpha\right)\) quá đơn giản

Giải \(\left(\beta\right)\) 

\(2\left(3sin\dfrac{x}{2}-4sin^3\dfrac{x}{x}\right)+cos\dfrac{x}{2}=0\)

⇔ \(-8sin^3\dfrac{x}{2}+6sin\dfrac{x}{2}\left(sin^2\dfrac{x}{2}+cos^2\dfrac{x}{2}\right)+cos\dfrac{x}{2}.\left(sin^2\dfrac{x}{2}+cos^2\dfrac{x}{2}\right)=0\)

⇔ \(-2sin^3\dfrac{x}{2}+6sin\dfrac{x}{2}.cos^2\dfrac{x}{2}+sin^2\dfrac{x}{2}.cos\dfrac{x}{2}+cos^3\dfrac{x}{2}=0\) 

Xét \(x=k2\pi,k\in Z\) tức \(sin\dfrac{x}{2}=0\) có thỏa mãn phương trình không, nếu có kết luận về nghiệm 

Dù trường hợp trên có thỏa mãn hay không thì tiếp tục xét trường hợp nữa là \(x\ne k2\pi,k\in Z\) tức \(sin\dfrac{x}{2}\ne0\). Rồi chia cả 2 vế phương trình lằng nhằng kia cho \(sin\dfrac{x}{2}\) và đưa về phương trình bậc 3 theo cot\(\dfrac{x}{2}\)

Nếu tham khảo theo cách của mình thì dùng công thức này : 

sin3x

= sin2x . cosx + cos2x . sinx 

= 2sinx . cosx . cosx + (1 - 2sin²x) . sinx

= 2sinx . cos²x + sinx - 2sin³x

= 2sinx (1 - sin²x) + sinx - 2sin³x

= 3sinx - 4sin³x