zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

sin(2x)=2sqrt(3)sin^2(x)

解答

sin (2 x) = 23 sin^{2} (x)

解答

x = 2 πn, x = π + 2 πn, x = \frac{π}{6} + πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 18 0^{\circ} n

求解步骤

sin (2 x) = 23 sin^{2} (x)

两边减去

23 sin^{2} (x)

sin (2 x) - 23 sin^{2} (x) = 0

使用三角恒等式改写

sin (2 x) - 2 sin^{2} (x) 3

使用倍角公式:

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

= 2 sin (x) cos (x) - 2 sin^{2} (x) 3

2 cos (x) sin (x) - 2 sin^{2} (x) 3 = 0

分解

2 cos (x) sin (x) - 2 sin^{2} (x) 3 : 2 sin (x) (cos (x) - 3 sin (x))

2 cos (x) sin (x) - 2 sin^{2} (x) 3

使用指数法则:

a^{b + c} = a^{b} a^{c}

sin^{2} (x) = sin (x) sin (x)

= 2 cos (x) sin (x) - 2 sin (x) sin (x) 3

改写为

= 2 sin (x) cos (x) - 2 sin (x) sin (x) 3

因式分解出通项

2 sin (x)

= 2 sin (x) (cos (x) - sin (x) 3)

2 sin (x) (cos (x) - 3 sin (x)) = 0

分别求解每个部分

sin (x) = 0 or cos (x) - 3 sin (x) = 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

cos (x) - 3 sin (x) = 0 : x = \frac{π}{6} + πn

cos (x) - 3 sin (x) = 0

使用三角恒等式改写

cos (x) - 3 sin (x) = 0

在两边除以

cos (x), cos (x) \neq = 0

\frac{cos ( x ) - 3 sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

化简

1 - \frac{3 sin ( x )}{cos ( x )} = 0

使用基本三角恒等式:

\frac{sin ( x )}{cos ( x )} = tan (x)

1 - 3 tan (x) = 0

1 - 3 tan (x) = 0

将

1

到右边

1 - 3 tan (x) = 0

两边减去

1

1 - 3 tan (x) - 1 = 0 - 1

化简

- 3 tan (x) = - 1

- 3 tan (x) = - 1

两边除以

- 3

- 3 tan (x) = - 1

两边除以

- 3

\frac{- 3 tan ( x )}{- 3} = \frac{- 1}{- 3}

化简

\frac{- 3 tan ( x )}{- 3} = \frac{- 1}{- 3}

化简

\frac{- 3 tan ( x )}{- 3} : tan (x)

\frac{- 3 tan ( x )}{- 3}

使用分式法则:

\frac{- a}{- b} = \frac{a}{b}

= \frac{3 tan ( x )}{3}

约分:

3

= tan (x)

化简

\frac{- 1}{- 3} : \frac{3}{3}

\frac{- 1}{- 3}

使用分式法则:

\frac{- a}{- b} = \frac{a}{b}

= \frac{1}{3}

\frac{1}{3}

有理化:

\frac{3}{3}

\frac{1}{3}

乘以共轭根式

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= \frac{3}{3}

= \frac{3}{3}

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3}

的通解

tan (x)

周期表（周期为

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

合并所有解

x = 2 πn, x = π + 2 πn, x = \frac{π}{6} + πn

作图

流行的例子

tan^2(x)+4tan(x)-5=0

3sin^2(x)-6sin(x)=0