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受欢迎的三角函数 >

sin(x)= 8/17 sin(2x)

解答

sin (x) = \frac{8}{17} sin (2 x)

解答

x = 2 πn, x = π + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

求解步骤

sin (x) = \frac{8}{17} sin (2 x)

两边减去

\frac{8}{17} sin (2 x)

sin (x) - \frac{8}{17} sin (2 x) = 0

化简

sin (x) - \frac{8}{17} sin (2 x) : \frac{17 sin ( x ) - 8 sin ( 2 x )}{17}

sin (x) - \frac{8}{17} sin (2 x)

乘

\frac{8}{17} sin (2 x) : \frac{8 sin ( 2 x )}{17}

\frac{8}{17} sin (2 x)

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{8 sin ( 2 x )}{17}

= sin (x) - \frac{8 sin ( 2 x )}{17}

将项转换为分式:

sin (x) = \frac{sin ( x ) 17}{17}

= \frac{sin ( x ) \cdot 17}{17} - \frac{8 sin ( 2 x )}{17}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{sin ( x ) \cdot 17 - 8 sin ( 2 x )}{17}

\frac{17 sin ( x ) - 8 sin ( 2 x )}{17} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

17 sin (x) - 8 sin (2 x) = 0

使用三角恒等式改写

17 sin (x) - 8 sin (2 x)

使用倍角公式:

sin (2 x) = 2 sin (x) cos (x)

= 17 sin (x) - 8 \cdot 2 sin (x) cos (x)

化简

= 17 sin (x) - 16 sin (x) cos (x)

17 sin (x) - 16 cos (x) sin (x) = 0

分解

17 sin (x) - 16 cos (x) sin (x) : - sin (x) (16 cos (x) - 17)

17 sin (x) - 16 cos (x) sin (x)

因式分解出通项

- sin (x)

= - sin (x) (- 17 + 16 cos (x))

- sin (x) (16 cos (x) - 17) = 0

分别求解每个部分

sin (x) = 0 or 16 cos (x) - 17 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (x) = 0

的通解

sin (x)

周期表（周期为

2 πn

"）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

16 cos (x) - 17 = 0 :

无解

16 cos (x) - 17 = 0

将

17

到右边

16 cos (x) - 17 = 0

两边加上

17

16 cos (x) - 17 + 17 = 0 + 17

化简

16 cos (x) = 17

16 cos (x) = 17

两边除以

16

16 cos (x) = 17

两边除以

16

\frac{16 cos ( x )}{16} = \frac{17}{16}

化简

cos (x) = \frac{17}{16}

cos (x) = \frac{17}{16}

- 1 \leq cos (x) \leq 1

无解

合并所有解

x = 2 πn, x = π + 2 πn

作图

流行的例子

tan(x)=-1,0<x<2pi

sec^2(x)-2=tan^2(x),0<= x<= 2pi