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受欢迎的三角函数 >

2cos^2(x/3)+3sin(x/3)-3=0

解答

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

解答

x = \frac{π}{2} + 6 πn, x = \frac{5 π}{2} + 6 πn, x = \frac{3 π}{2} + 6 πn

+1

度数

x = 9 0^{\circ} + 108 0^{\circ} n, x = 45 0^{\circ} + 108 0^{\circ} n, x = 27 0^{\circ} + 108 0^{\circ} n

求解步骤

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

使用三角恒等式改写

- 3 + 2 cos^{2} (\frac{x}{3}) + 3 sin (\frac{x}{3})

使用毕达哥拉斯恒等式:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= - 3 + 2 (1 - sin^{2} (\frac{x}{3})) + 3 sin (\frac{x}{3})

化简

- 3 + 2 (1 - sin^{2} (\frac{x}{3})) + 3 sin (\frac{x}{3}) : 3 sin (\frac{x}{3}) - 2 sin^{2} (\frac{x}{3}) - 1

- 3 + 2 (1 - sin^{2} (\frac{x}{3})) + 3 sin (\frac{x}{3})

乘开

2 (1 - sin^{2} (\frac{x}{3})) : 2 - 2 sin^{2} (\frac{x}{3})

2 (1 - sin^{2} (\frac{x}{3}))

使用分配律:

a (b - c) = ab - a c

a = 2, b = 1, c = sin^{2} (\frac{x}{3})

= 2 \cdot 1 - 2 sin^{2} (\frac{x}{3})

数字相乘:

2 \cdot 1 = 2

= 2 - 2 sin^{2} (\frac{x}{3})

= - 3 + 2 - 2 sin^{2} (\frac{x}{3}) + 3 sin (\frac{x}{3})

数字相加/相减:

- 3 + 2 = - 1

= 3 sin (\frac{x}{3}) - 2 sin^{2} (\frac{x}{3}) - 1

= 3 sin (\frac{x}{3}) - 2 sin^{2} (\frac{x}{3}) - 1

- 1 - 2 sin^{2} (\frac{x}{3}) + 3 sin (\frac{x}{3}) = 0

用替代法求解

- 1 - 2 sin^{2} (\frac{x}{3}) + 3 sin (\frac{x}{3}) = 0

令

: sin (\frac{x}{3}) = u

- 1 - 2 u^{2} + 3 u = 0

- 1 - 2 u^{2} + 3 u = 0 : u = \frac{1}{2}, u = 1

- 1 - 2 u^{2} + 3 u = 0

改写成标准形式

a x^{2} + b x + c = 0

- 2 u^{2} + 3 u - 1 = 0

使用求根公式求解

- 2 u^{2} + 3 u - 1 = 0

二次方程求根公式:

若

a = - 2, b = 3, c = - 1

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 2 ) ( - 1 )}{2 ( - 2 )}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 2 ) ( - 1 )}{2 ( - 2 )}

3^{2} - 4 (- 2) (- 1) = 1

3^{2} - 4 (- 2) (- 1)

使用法则

- (- a) = a

= 3^{2} - 4 \cdot 2 \cdot 1

数字相乘:

4 \cdot 2 \cdot 1 = 8

= 3^{2} - 8

3^{2} = 9

= 9 - 8

数字相减:

9 - 8 = 1

= 1

使用法则

1 = 1

= 1

u_{1, 2} = \frac{- 3 \pm 1}{2 ( - 2 )}

将解分隔开

u_{1} = \frac{- 3 + 1}{2 ( - 2 )}, u_{2} = \frac{- 3 - 1}{2 ( - 2 )}

u = \frac{- 3 + 1}{2 ( - 2 )} : \frac{1}{2}

\frac{- 3 + 1}{2 ( - 2 )}

去除括号:

(- a) = - a

= \frac{- 3 + 1}{- 2 \cdot 2}

数字相加/相减:

- 3 + 1 = - 2

= \frac{- 2}{- 2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 2}{- 4}

使用分式法则:

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

= \frac{2}{4}

约分:

2

= \frac{1}{2}

u = \frac{- 3 - 1}{2 ( - 2 )} : 1

\frac{- 3 - 1}{2 ( - 2 )}

去除括号:

(- a) = - a

= \frac{- 3 - 1}{- 2 \cdot 2}

数字相减:

- 3 - 1 = - 4

= \frac{- 4}{- 2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 4}{- 4}

使用分式法则:

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

= \frac{4}{4}

使用法则

\frac{a}{a} = 1

= 1

二次方程组的解是:

u = \frac{1}{2}, u = 1

u = sin (\frac{x}{3})

代回

sin (\frac{x}{3}) = \frac{1}{2}, sin (\frac{x}{3}) = 1

sin (\frac{x}{3}) = \frac{1}{2}, sin (\frac{x}{3}) = 1

sin (\frac{x}{3}) = \frac{1}{2} : x = \frac{π}{2} + 6 πn, x = \frac{5 π}{2} + 6 πn

sin (\frac{x}{3}) = \frac{1}{2}

sin (\frac{x}{3}) = \frac{1}{2}

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}

\frac{x}{3} = \frac{π}{6} + 2 πn, \frac{x}{3} = \frac{5 π}{6} + 2 πn

\frac{x}{3} = \frac{π}{6} + 2 πn, \frac{x}{3} = \frac{5 π}{6} + 2 πn

解

\frac{x}{3} = \frac{π}{6} + 2 πn : x = \frac{π}{2} + 6 πn

\frac{x}{3} = \frac{π}{6} + 2 πn

在两边乘以

3

\frac{x}{3} = \frac{π}{6} + 2 πn

在两边乘以

3

\frac{3 x}{3} = 3 \cdot \frac{π}{6} + 3 \cdot 2 πn

化简

\frac{3 x}{3} = 3 \cdot \frac{π}{6} + 3 \cdot 2 πn

化简

\frac{3 x}{3} : x

\frac{3 x}{3}

数字相除:

\frac{3}{3} = 1

= x

化简

3 \cdot \frac{π}{6} + 3 \cdot 2 πn : \frac{π}{2} + 6 πn

3 \cdot \frac{π}{6} + 3 \cdot 2 πn

3 \cdot \frac{π}{6} = \frac{π}{2}

3 \cdot \frac{π}{6}

分式相乘:

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

= \frac{π 3}{6}

约分:

3

= \frac{π}{2}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

数字相乘:

3 \cdot 2 = 6

= 6 πn

= \frac{π}{2} + 6 πn

x = \frac{π}{2} + 6 πn

x = \frac{π}{2} + 6 πn

x = \frac{π}{2} + 6 πn

解

\frac{x}{3} = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{2} + 6 πn

\frac{x}{3} = \frac{5 π}{6} + 2 πn

在两边乘以

3

\frac{x}{3} = \frac{5 π}{6} + 2 πn

在两边乘以

3

\frac{3 x}{3} = 3 \cdot \frac{5 π}{6} + 3 \cdot 2 πn

化简

\frac{3 x}{3} = 3 \cdot \frac{5 π}{6} + 3 \cdot 2 πn

化简

\frac{3 x}{3} : x

\frac{3 x}{3}

数字相除:

\frac{3}{3} = 1

= x

化简

3 \cdot \frac{5 π}{6} + 3 \cdot 2 πn : \frac{5 π}{2} + 6 πn

3 \cdot \frac{5 π}{6} + 3 \cdot 2 πn

3 \cdot \frac{5 π}{6} = \frac{5 π}{2}

3 \cdot \frac{5 π}{6}

分式相乘:

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

= \frac{5 π 3}{6}

数字相乘:

5 \cdot 3 = 15

= \frac{15 π}{6}

约分:

3

= \frac{5 π}{2}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

数字相乘:

3 \cdot 2 = 6

= 6 πn

= \frac{5 π}{2} + 6 πn

x = \frac{5 π}{2} + 6 πn

x = \frac{5 π}{2} + 6 πn

x = \frac{5 π}{2} + 6 πn

x = \frac{π}{2} + 6 πn, x = \frac{5 π}{2} + 6 πn

sin (\frac{x}{3}) = 1 : x = \frac{3 π}{2} + 6 πn

sin (\frac{x}{3}) = 1

sin (\frac{x}{3}) = 1

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}

\frac{x}{3} = \frac{π}{2} + 2 πn

\frac{x}{3} = \frac{π}{2} + 2 πn

解

\frac{x}{3} = \frac{π}{2} + 2 πn : x = \frac{3 π}{2} + 6 πn

\frac{x}{3} = \frac{π}{2} + 2 πn

在两边乘以

3

\frac{x}{3} = \frac{π}{2} + 2 πn

在两边乘以

3

\frac{3 x}{3} = 3 \cdot \frac{π}{2} + 3 \cdot 2 πn

化简

\frac{3 x}{3} = 3 \cdot \frac{π}{2} + 3 \cdot 2 πn

化简

\frac{3 x}{3} : x

\frac{3 x}{3}

数字相除:

\frac{3}{3} = 1

= x

化简

3 \cdot \frac{π}{2} + 3 \cdot 2 πn : \frac{3 π}{2} + 6 πn

3 \cdot \frac{π}{2} + 3 \cdot 2 πn

乘

3 \cdot \frac{π}{2} : \frac{3 π}{2}

3 \cdot \frac{π}{2}

分式相乘:

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

= \frac{π 3}{2}

= \frac{3 π}{2} + 3 \cdot 2 πn

数字相乘:

3 \cdot 2 = 6

= \frac{3 π}{2} + 6 πn

x = \frac{3 π}{2} + 6 πn

x = \frac{3 π}{2} + 6 πn

x = \frac{3 π}{2} + 6 πn

x = \frac{3 π}{2} + 6 πn

合并所有解

x = \frac{π}{2} + 6 πn, x = \frac{5 π}{2} + 6 πn, x = \frac{3 π}{2} + 6 πn

作图

流行的例子

tan(2θ)+tan(θ)=0

tan (2 θ) + tan (θ) = 0

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

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

4cos(x)=2sqrt(3)

4 cos (x) = 23

tan (x) = \frac{50}{50}

4 tan (θ) = 4