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受欢迎的三角函数 >

15cos^2(x)+7cos(x)-2=0

解答

15 cos^{2} (x) + 7 cos (x) - 2 = 0

解答

x = 1.36943 \dots + 2 πn, x = 2 π - 1.36943 \dots + 2 πn, x = 2.30052 \dots + 2 πn, x = - 2.30052 \dots + 2 πn

+1

度数

x = 78.46304 \dots^{\circ} + 36 0^{\circ} n, x = 281.53695 \dots^{\circ} + 36 0^{\circ} n, x = 131.81031 \dots^{\circ} + 36 0^{\circ} n, x = - 131.81031 \dots^{\circ} + 36 0^{\circ} n

求解步骤

15 cos^{2} (x) + 7 cos (x) - 2 = 0

用替代法求解

15 cos^{2} (x) + 7 cos (x) - 2 = 0

令

: cos (x) = u

15 u^{2} + 7 u - 2 = 0

15 u^{2} + 7 u - 2 = 0 : u = \frac{1}{5}, u = - \frac{2}{3}

15 u^{2} + 7 u - 2 = 0

使用求根公式求解

15 u^{2} + 7 u - 2 = 0

二次方程求根公式:

若

a = 15, b = 7, c = - 2

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 \cdot 15 ( - 2 )}{2 \cdot 15}

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 \cdot 15 ( - 2 )}{2 \cdot 15}

7^{2} - 4 \cdot 15 (- 2) = 13

7^{2} - 4 \cdot 15 (- 2)

使用法则

- (- a) = a

= 7^{2} + 4 \cdot 15 \cdot 2

数字相乘:

4 \cdot 15 \cdot 2 = 120

= 7^{2} + 120

7^{2} = 49

= 49 + 120

数字相加:

49 + 120 = 169

= 169

因式分解数字:

169 = 1 3^{2}

= 1 3^{2}

使用根式运算法则:

n a^{n} = a

1 3^{2} = 13

= 13

u_{1, 2} = \frac{- 7 \pm 13}{2 \cdot 15}

将解分隔开

u_{1} = \frac{- 7 + 13}{2 \cdot 15}, u_{2} = \frac{- 7 - 13}{2 \cdot 15}

u = \frac{- 7 + 13}{2 \cdot 15} : \frac{1}{5}

\frac{- 7 + 13}{2 \cdot 15}

数字相加/相减:

- 7 + 13 = 6

= \frac{6}{2 \cdot 15}

数字相乘:

2 \cdot 15 = 30

= \frac{6}{30}

约分:

6

= \frac{1}{5}

u = \frac{- 7 - 13}{2 \cdot 15} : - \frac{2}{3}

\frac{- 7 - 13}{2 \cdot 15}

数字相减:

- 7 - 13 = - 20

= \frac{- 20}{2 \cdot 15}

数字相乘:

2 \cdot 15 = 30

= \frac{- 20}{30}

使用分式法则:

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

= - \frac{20}{30}

约分:

10

= - \frac{2}{3}

二次方程组的解是:

u = \frac{1}{5}, u = - \frac{2}{3}

u = cos (x)

代回

cos (x) = \frac{1}{5}, cos (x) = - \frac{2}{3}

cos (x) = \frac{1}{5}, cos (x) = - \frac{2}{3}

cos (x) = \frac{1}{5} : x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn

cos (x) = \frac{1}{5}

使用反三角函数性质

cos (x) = \frac{1}{5}

cos (x) = \frac{1}{5}

的通解

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

x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn

x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn

cos (x) = - \frac{2}{3} : x = arccos (- \frac{2}{3}) + 2 πn, x = - arccos (- \frac{2}{3}) + 2 πn

cos (x) = - \frac{2}{3}

使用反三角函数性质

cos (x) = - \frac{2}{3}

cos (x) = - \frac{2}{3}

的通解

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

x = arccos (- \frac{2}{3}) + 2 πn, x = - arccos (- \frac{2}{3}) + 2 πn

x = arccos (- \frac{2}{3}) + 2 πn, x = - arccos (- \frac{2}{3}) + 2 πn

合并所有解

x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn, x = arccos (- \frac{2}{3}) + 2 πn, x = - arccos (- \frac{2}{3}) + 2 πn

以小数形式表示解

x = 1.36943 \dots + 2 πn, x = 2 π - 1.36943 \dots + 2 πn, x = 2.30052 \dots + 2 πn, x = - 2.30052 \dots + 2 πn

作图

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