Question

1. Sinbìnhx + sinbình3x = cosbìnhx + cosbình3x

2. Cos(pi/3+x) + cos(pi/3-x) = 1

3. Các nghiệm thuộc khoảng (0;2pi) của: sinmũbốnx/2 + cosmũbốnx/2 = 5/8

4. Sinx + cosx + √2 sin2x = 0

5. Số nghiệm thuộc khoảng [0;14] của: cos3x - 4cos2x + 3cosx - 4 =0

Answer 1

Sử dụng bộ gõ công thức ở chỗ khoanh đỏ trên khung soạn thảo cho người khác dễ đọc đi bạn:

1.

\(sin^2x+sin^23x=cos^2x+cos^23x\)

\(\Leftrightarrow sin^2x-cos^2x=cos^23x-sin^23x\)

\(\Leftrightarrow-cos2x=cos6x\)

\(\Leftrightarrow cos2x+cos6x=0\)

\(\Leftrightarrow2cos4x.cos2x=0\)

\(\Rightarrow\left[{}\begin{matrix}cos4x=0\\cos2x=0\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}4x=\frac{\pi}{2}+k\pi\\2x=\frac{\pi}{2}+k\pi\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=\frac{\pi}{8}+\frac{k\pi}{4}\\x=\frac{\pi}{4}+\frac{k\pi}{2}\end{matrix}\right.\)

Answer 2

2.

\(cos\left(\frac{\pi}{3}+x\right)+cos\left(\frac{\pi}{3}-x\right)=1\)

\(\Leftrightarrow2cos\left(\frac{\pi}{3}\right).cosx=1\)

\(\Leftrightarrow cosx=1\)

\(\Rightarrow x=k2\pi\)

3.

\(sinx+cosx+\sqrt{2}sin2x=0\)

\(\Leftrightarrow\sqrt{2}sin\left(x+\frac{\pi}{4}\right)+\sqrt{2}sin2x=0\)

\(\Leftrightarrow sin\left(x+\frac{\pi}{4}\right)=-sin2x=sin\left(-2x\right)\)

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

\(\Rightarrow\left[{}\begin{matrix}x=-\frac{\pi}{12}+\frac{k2\pi}{3}\\x=-\frac{3\pi}{4}+k2\pi\end{matrix}\right.\)

Answer 3

4.

\(sin^4\frac{x}{2}+cos^4\frac{x}{2}=\frac{5}{8}\)

\(\Leftrightarrow\left(sin^2\frac{x}{2}+cos^2\frac{x}{2}\right)^2-2sin^2\frac{x}{2}.cos^2\frac{x}{2}=\frac{5}{8}\)

\(\Leftrightarrow1-\frac{1}{2}\left(2sin\frac{x}{2}.cos\frac{x}{2}\right)^2=\frac{5}{8}\)

\(\Leftrightarrow1-\frac{1}{2}sin^2x=\frac{5}{8}\)

\(\Leftrightarrow sin^2x=\frac{3}{4}\Rightarrow\left[{}\begin{matrix}sinx=\frac{\sqrt{3}}{2}\\sinx=-\frac{\sqrt{3}}{2}\end{matrix}\right.\)

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

\(\Rightarrow x=\left\{\frac{\pi}{3};\frac{2\pi}{3};\frac{5\pi}{3};\frac{4\pi}{3}\right\}\)

Answer 4

5.

\(cos3x-4cos2x+3cosx-4=0\)

\(\Leftrightarrow4cos^3x-3cosx-4\left(2cos^2x-1\right)+3cosx-4=0\)

\(\Leftrightarrow4cos^3x-8cos^2x=0\)

\(\Leftrightarrow4cos^2x\left(cosx-2\right)=0\)

\(\Leftrightarrow cosx=0\)

\(\Leftrightarrow x=\frac{\pi}{2}+k\pi\)

\(0\le x\le14\Rightarrow0\le\frac{\pi}{2}+k\pi\le14\)

\(\Rightarrow-\frac{1}{2}\le k\le\frac{28-\pi}{2\pi}\)

Mà k nguyên \(\Rightarrow k=\left\{0;1;2;3\right\}\)

\(\Rightarrow x=\left\{\frac{\pi}{2};\frac{3\pi}{2};\frac{5\pi}{2};\frac{7\pi}{2}\right\}\)