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受欢迎的三角函数 >

7cos^2(x)-1=5cos(x)

解答

7 cos^{2} (x) - 1 = 5 cos (x)

解答

x = 0.50090 \dots + 2 πn, x = 2 π - 0.50090 \dots + 2 πn, x = 1.73439 \dots + 2 πn, x = - 1.73439 \dots + 2 πn

+1

度数

x = 28.69945 \dots^{\circ} + 36 0^{\circ} n, x = 331.30054 \dots^{\circ} + 36 0^{\circ} n, x = 99.37322 \dots^{\circ} + 36 0^{\circ} n, x = - 99.37322 \dots^{\circ} + 36 0^{\circ} n

求解步骤

7 cos^{2} (x) - 1 = 5 cos (x)

用替代法求解

7 cos^{2} (x) - 1 = 5 cos (x)

令

: cos (x) = u

7 u^{2} - 1 = 5 u

7 u^{2} - 1 = 5 u : u = \frac{5 + 53}{14}, u = \frac{5 - 53}{14}

7 u^{2} - 1 = 5 u

将

5 u

para o lado esquerdo

7 u^{2} - 1 = 5 u

两边减去

5 u

7 u^{2} - 1 - 5 u = 5 u - 5 u

化简

7 u^{2} - 1 - 5 u = 0

7 u^{2} - 1 - 5 u = 0

改写成标准形式

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

7 u^{2} - 5 u - 1 = 0

使用求根公式求解

7 u^{2} - 5 u - 1 = 0

二次方程求根公式:

若

a = 7, b = - 5, c = - 1

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 7 ( - 1 )}{2 \cdot 7}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 7 ( - 1 )}{2 \cdot 7}

(- 5)^{2} - 4 \cdot 7 (- 1) = 53

(- 5)^{2} - 4 \cdot 7 (- 1)

使用法则

- (- a) = a

= (- 5)^{2} + 4 \cdot 7 \cdot 1

使用指数法则:

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

若

n

是偶数

(- 5)^{2} = 5^{2}

= 5^{2} + 4 \cdot 7 \cdot 1

数字相乘:

4 \cdot 7 \cdot 1 = 28

= 5^{2} + 28

5^{2} = 25

= 25 + 28

数字相加:

25 + 28 = 53

= 53

u_{1, 2} = \frac{- ( - 5 ) \pm 53}{2 \cdot 7}

将解分隔开

u_{1} = \frac{- ( - 5 ) + 53}{2 \cdot 7}, u_{2} = \frac{- ( - 5 ) - 53}{2 \cdot 7}

u = \frac{- ( - 5 ) + 53}{2 \cdot 7} : \frac{5 + 53}{14}

\frac{- ( - 5 ) + 53}{2 \cdot 7}

使用法则

- (- a) = a

= \frac{5 + 53}{2 \cdot 7}

数字相乘:

2 \cdot 7 = 14

= \frac{5 + 53}{14}

u = \frac{- ( - 5 ) - 53}{2 \cdot 7} : \frac{5 - 53}{14}

\frac{- ( - 5 ) - 53}{2 \cdot 7}

使用法则

- (- a) = a

= \frac{5 - 53}{2 \cdot 7}

数字相乘:

2 \cdot 7 = 14

= \frac{5 - 53}{14}

二次方程组的解是:

u = \frac{5 + 53}{14}, u = \frac{5 - 53}{14}

u = cos (x)

代回

cos (x) = \frac{5 + 53}{14}, cos (x) = \frac{5 - 53}{14}

cos (x) = \frac{5 + 53}{14}, cos (x) = \frac{5 - 53}{14}

cos (x) = \frac{5 + 53}{14} : x = arccos (\frac{5 + 53}{14}) + 2 πn, x = 2 π - arccos (\frac{5 + 53}{14}) + 2 πn

cos (x) = \frac{5 + 53}{14}

使用反三角函数性质

cos (x) = \frac{5 + 53}{14}

cos (x) = \frac{5 + 53}{14}

的通解

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

x = arccos (\frac{5 + 53}{14}) + 2 πn, x = 2 π - arccos (\frac{5 + 53}{14}) + 2 πn

x = arccos (\frac{5 + 53}{14}) + 2 πn, x = 2 π - arccos (\frac{5 + 53}{14}) + 2 πn

cos (x) = \frac{5 - 53}{14} : x = arccos (\frac{5 - 53}{14}) + 2 πn, x = - arccos (\frac{5 - 53}{14}) + 2 πn

cos (x) = \frac{5 - 53}{14}

使用反三角函数性质

cos (x) = \frac{5 - 53}{14}

cos (x) = \frac{5 - 53}{14}

的通解

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

x = arccos (\frac{5 - 53}{14}) + 2 πn, x = - arccos (\frac{5 - 53}{14}) + 2 πn

x = arccos (\frac{5 - 53}{14}) + 2 πn, x = - arccos (\frac{5 - 53}{14}) + 2 πn

合并所有解

x = arccos (\frac{5 + 53}{14}) + 2 πn, x = 2 π - arccos (\frac{5 + 53}{14}) + 2 πn, x = arccos (\frac{5 - 53}{14}) + 2 πn, x = - arccos (\frac{5 - 53}{14}) + 2 πn

以小数形式表示解

x = 0.50090 \dots + 2 πn, x = 2 π - 0.50090 \dots + 2 πn, x = 1.73439 \dots + 2 πn, x = - 1.73439 \dots + 2 πn

作图

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