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受欢迎的三角函数 >

2sin(x)=sqrt(cos(2x)+2)

解答

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

解答

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

+1

度数

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

cos (2 x) + 2

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

使用三角恒等式改写

- 2 + cos (2 x) + 2 sin (x)

使用倍角公式:

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

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

化简

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

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

用替代法求解

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

令

: sin (x) = u

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

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

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

去除平方根

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

两边减去

2 u

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

化简

- 3 - 2 u^{2} = - 2 u

两边进行平方:

3 - 2 u^{2} = 4 u^{2}

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

(- 3 - 2 u^{2})^{2} = (- 2 u)^{2}

展开

(- 3 - 2 u^{2})^{2} : 3 - 2 u^{2}

(- 3 - 2 u^{2})^{2}

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- 3 - 2 u^{2})^{2} = (3 - 2 u^{2})^{2}

= (3 - 2 u^{2})^{2}

使用根式运算法则:

a = a^{\frac{1}{2}}

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

使用指数法则:

(a^{b})^{c} = a^{b c}

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

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

\frac{1}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 1

= 3 - 2 u^{2}

展开

(- 2 u)^{2} : 4 u^{2}

(- 2 u)^{2}

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- 2 u)^{2} = (2 u)^{2}

= (2 u)^{2}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} u^{2}

2^{2} = 4

= 4 u^{2}

3 - 2 u^{2} = 4 u^{2}

3 - 2 u^{2} = 4 u^{2}

3 - 2 u^{2} = 4 u^{2}

解

3 - 2 u^{2} = 4 u^{2} : u = \frac{1}{2}, u = - \frac{1}{2}

3 - 2 u^{2} = 4 u^{2}

将

3

到右边

3 - 2 u^{2} = 4 u^{2}

两边减去

3

3 - 2 u^{2} - 3 = 4 u^{2} - 3

化简

- 2 u^{2} = 4 u^{2} - 3

- 2 u^{2} = 4 u^{2} - 3

将

4 u^{2}

para o lado esquerdo

- 2 u^{2} = 4 u^{2} - 3

两边减去

4 u^{2}

- 2 u^{2} - 4 u^{2} = 4 u^{2} - 3 - 4 u^{2}

化简

- 6 u^{2} = - 3

- 6 u^{2} = - 3

两边除以

- 6

- 6 u^{2} = - 3

两边除以

- 6

\frac{- 6 u ^{2}}{- 6} = \frac{- 3}{- 6}

化简

\frac{- 6 u ^{2}}{- 6} = \frac{- 3}{- 6}

化简

\frac{- 6 u ^{2}}{- 6} : u^{2}

\frac{- 6 u ^{2}}{- 6}

使用分式法则:

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

= \frac{6 u ^{2}}{6}

数字相除:

\frac{6}{6} = 1

= u^{2}

化简

\frac{- 3}{- 6} : \frac{1}{2}

\frac{- 3}{- 6}

使用分式法则:

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

= \frac{3}{6}

约分:

3

= \frac{1}{2}

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

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

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

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

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

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

验证解:

u = \frac{1}{2}

真

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

假

将它们代入

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

检验解是否符合

去除与方程不符的解。

代入

u = \frac{1}{2} :

真

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

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

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

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

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

2 (\frac{1}{2})^{2} = 1

2 (\frac{1}{2})^{2}

(\frac{1}{2})^{2} = \frac{1}{2}

(\frac{1}{2})^{2}

使用根式运算法则:

a = a^{\frac{1}{2}}

= ((\frac{1}{2})^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= (\frac{1}{2})^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 1

= \frac{1}{2}

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

分式相乘:

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

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

约分:

2

= 1

= 3 - 1

数字相减:

3 - 1 = 2

= 2

2 \frac{1}{2} = 2

2 \frac{1}{2}

使用指数法则:

a \frac{b}{a} = a^{2} \frac{b}{a}

2 \frac{1}{2} = 2^{2} \cdot \frac{1}{2}

= 2^{2} \cdot \frac{1}{2}

乘

2^{2} \cdot \frac{1}{2} : 2

2^{2} \cdot \frac{1}{2}

分式相乘:

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

= \frac{1 \cdot 2 ^{2}}{2}

乘以:

1 \cdot 2^{2} = 2^{2}

= \frac{2 ^{2}}{2}

约分:

2

= 2

= 2

= - 2 + 2

同类项相加:

- 2 + 2 = 0

= 0

0 = 0

真

代入

u = - \frac{1}{2} :

假

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

- 3 - 2 (- \frac{1}{2})^{2} + 2 (- \frac{1}{2}) = - 22

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

去除括号:

(- a) = - a

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

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

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

2 (- \frac{1}{2})^{2} = 1

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

(- \frac{1}{2})^{2} = \frac{1}{2}

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

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- \frac{1}{2})^{2} = (\frac{1}{2})^{2}

= (\frac{1}{2})^{2}

使用根式运算法则:

a = a^{\frac{1}{2}}

= ((\frac{1}{2})^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= (\frac{1}{2})^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 1

= \frac{1}{2}

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

分式相乘:

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

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

约分:

2

= 1

= 3 - 1

数字相减:

3 - 1 = 2

= 2

2 \frac{1}{2} = 2

2 \frac{1}{2}

使用指数法则:

a \frac{b}{a} = a^{2} \frac{b}{a}

2 \frac{1}{2} = 2^{2} \cdot \frac{1}{2}

= 2^{2} \cdot \frac{1}{2}

乘

2^{2} \cdot \frac{1}{2} : 2

2^{2} \cdot \frac{1}{2}

分式相乘:

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

= \frac{1 \cdot 2 ^{2}}{2}

乘以:

1 \cdot 2^{2} = 2^{2}

= \frac{2 ^{2}}{2}

约分:

2

= 2

= 2

= - 2 - 2

同类项相加:

- 2 - 2 = - 22

= - 22

- 22 = 0

假

解是

u = \frac{1}{2}

u = sin (x)

代回

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

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

sin (x) = \frac{1}{2} : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

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

sin (x) = \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}

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

合并所有解

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

作图

流行的例子

tan (x) = \frac{7}{12}

cos(x)=sqrt(1-sin(x))

cos (x) = 1 - sin (x)

-sin(a)-1=3sin(a)+2

- sin (a) - 1 = 3 sin (a) + 2

tan (x) = 7

cot(x)(tan(x)-sqrt(3))=0

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