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cos^4(x)=0.37

解答

cos^{4} (x) = 0.37

解答

x = 0.67625 \dots + 2 πn, x = 2 π - 0.67625 \dots + 2 πn, x = 2.46533 \dots + 2 πn, x = - 2.46533 \dots + 2 πn

+1

度数

x = 38.74668 \dots^{\circ} + 36 0^{\circ} n, x = 321.25331 \dots^{\circ} + 36 0^{\circ} n, x = 141.25331 \dots^{\circ} + 36 0^{\circ} n, x = - 141.25331 \dots^{\circ} + 36 0^{\circ} n

求解步骤

cos^{4} (x) = 0.37

用替代法求解

cos^{4} (x) = 0.37

令

: cos (x) = u

u^{4} = 0.37

u^{4} = 0.37

用

v = u^{2}

和

v^{2} = u^{4}

改写方程式

v^{2} = 0.37

解

v^{2} = 0.37 : v = 0.37, v = - 0.37

v^{2} = 0.37

对于

(g (x))^{2} = f (a)

解为

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

v = 0.37, v = - 0.37

v = 0.37, v = - 0.37

代回

v = u^{2}

，求解

u

解

u^{2} = 0.37

对于

x^{2} = f (a)

解为

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

u = 0.37, u = - 0.37

0.37

使用根式运算法则:

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

= (0.3 7^{\frac{1}{2}})^{\frac{1}{2}}

使用指数法则:

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

= 0.3 7^{\frac{1}{2} \cdot \frac{1}{2}}

\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}

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

分式相乘:

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

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

数字相乘:

1 \cdot 1 = 1

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

数字相乘:

2 \cdot 2 = 4

= \frac{1}{4}

= 0.3 7^{\frac{1}{4}}

- 0.37

化简

0.37

使用根式运算法则:

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

= (0.3 7^{\frac{1}{2}})^{\frac{1}{2}}

使用指数法则:

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

= 0.3 7^{\frac{1}{2} \cdot \frac{1}{2}}

\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}

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

分式相乘:

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

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

数字相乘:

1 \cdot 1 = 1

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

数字相乘:

2 \cdot 2 = 4

= \frac{1}{4}

= 0.3 7^{\frac{1}{4}}

解

u^{2} = - 0.37

对于

x^{2} = f (a)

解为

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

u = - 0.37, u = - - 0.37

化简

- 0.37

使用根式运算法则:

- a = - 1 a

- 0.37 = - 1 0.37

= - 1 0.37

使用虚数运算法则:

- 1 = i

= i 0.37

使用根式运算法则:

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

= (0.3 7^{\frac{1}{2}})^{\frac{1}{2}}

使用指数法则:

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

= 0.3 7^{\frac{1}{2} \cdot \frac{1}{2}}

\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}

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

分式相乘:

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

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

数字相乘:

1 \cdot 1 = 1

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

数字相乘:

2 \cdot 2 = 4

= \frac{1}{4}

= 0.3 7^{\frac{1}{4}}

化简

- - 0.37

化简

- 0.37

使用根式运算法则:

- a = - 1 a

- 0.37 = - 1 0.37

= - 1 0.37

使用虚数运算法则:

- 1 = i

= i 0.37

使用根式运算法则:

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

= (0.3 7^{\frac{1}{2}})^{\frac{1}{2}}

使用指数法则:

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

= 0.3 7^{\frac{1}{2} \cdot \frac{1}{2}}

\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}

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

分式相乘:

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

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

数字相乘:

1 \cdot 1 = 1

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

数字相乘:

2 \cdot 2 = 4

= \frac{1}{4}

= 0.3 7^{\frac{1}{4}}

解为

u = cos (x)

代回

使用反三角函数性质

的通解

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

使用反三角函数性质

的通解

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

无解

无解

无解

无解

合并所有解

以小数形式表示解

x = 0.67625 \dots + 2 πn, x = 2 π - 0.67625 \dots + 2 πn, x = 2.46533 \dots + 2 πn, x = - 2.46533 \dots + 2 πn

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