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21+18cos(x)=16(1-cos^2(x))

解答

21 + 18 cos (x) = 16 (1 - cos^{2} (x))

解答

x = 2.24592 \dots + 2 πn, x = - 2.24592 \dots + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

+1

度数

x = 128.68218 \dots^{\circ} + 36 0^{\circ} n, x = - 128.68218 \dots^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

求解步骤

21 + 18 cos (x) = 16 (1 - cos^{2} (x))

用替代法求解

21 + 18 cos (x) = 16 (1 - cos^{2} (x))

令

: cos (x) = u

21 + 18 u = 16 (1 - u^{2})

21 + 18 u = 16 (1 - u^{2}) : u = - \frac{5}{8}, u = - \frac{1}{2}

21 + 18 u = 16 (1 - u^{2})

展开

16 (1 - u^{2}) : 16 - 16 u^{2}

16 (1 - u^{2})

使用分配律:

a (b - c) = ab - a c

a = 16, b = 1, c = u^{2}

= 16 \cdot 1 - 16 u^{2}

数字相乘:

16 \cdot 1 = 16

= 16 - 16 u^{2}

21 + 18 u = 16 - 16 u^{2}

交换两边

16 - 16 u^{2} = 21 + 18 u

将

18 u

para o lado esquerdo

16 - 16 u^{2} = 21 + 18 u

两边减去

18 u

16 - 16 u^{2} - 18 u = 21 + 18 u - 18 u

化简

16 - 16 u^{2} - 18 u = 21

16 - 16 u^{2} - 18 u = 21

将

21

para o lado esquerdo

16 - 16 u^{2} - 18 u = 21

两边减去

21

16 - 16 u^{2} - 18 u - 21 = 21 - 21

化简

- 16 u^{2} - 18 u - 5 = 0

- 16 u^{2} - 18 u - 5 = 0

使用求根公式求解

- 16 u^{2} - 18 u - 5 = 0

二次方程求根公式:

若

a = - 16, b = - 18, c = - 5

u_{1, 2} = \frac{- ( - 18 ) \pm ( - 18 ) ^{2} - 4 ( - 16 ) ( - 5 )}{2 ( - 16 )}

u_{1, 2} = \frac{- ( - 18 ) \pm ( - 18 ) ^{2} - 4 ( - 16 ) ( - 5 )}{2 ( - 16 )}

(- 18)^{2} - 4 (- 16) (- 5) = 2

(- 18)^{2} - 4 (- 16) (- 5)

使用法则

- (- a) = a

= (- 18)^{2} - 4 \cdot 16 \cdot 5

使用指数法则:

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

若

n

是偶数

(- 18)^{2} = 1 8^{2}

= 1 8^{2} - 4 \cdot 16 \cdot 5

数字相乘:

4 \cdot 16 \cdot 5 = 320

= 1 8^{2} - 320

1 8^{2} = 324

= 324 - 320

数字相减:

324 - 320 = 4

= 4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

2^{2} = 2

= 2

u_{1, 2} = \frac{- ( - 18 ) \pm 2}{2 ( - 16 )}

将解分隔开

u_{1} = \frac{- ( - 18 ) + 2}{2 ( - 16 )}, u_{2} = \frac{- ( - 18 ) - 2}{2 ( - 16 )}

u = \frac{- ( - 18 ) + 2}{2 ( - 16 )} : - \frac{5}{8}

\frac{- ( - 18 ) + 2}{2 ( - 16 )}

去除括号:

(- a) = - a, - (- a) = a

= \frac{18 + 2}{- 2 \cdot 16}

数字相加:

18 + 2 = 20

= \frac{20}{- 2 \cdot 16}

数字相乘:

2 \cdot 16 = 32

= \frac{20}{- 32}

使用分式法则:

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

= - \frac{20}{32}

约分:

4

= - \frac{5}{8}

u = \frac{- ( - 18 ) - 2}{2 ( - 16 )} : - \frac{1}{2}

\frac{- ( - 18 ) - 2}{2 ( - 16 )}

去除括号:

(- a) = - a, - (- a) = a

= \frac{18 - 2}{- 2 \cdot 16}

数字相减:

18 - 2 = 16

= \frac{16}{- 2 \cdot 16}

数字相乘:

2 \cdot 16 = 32

= \frac{16}{- 32}

使用分式法则:

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

= - \frac{16}{32}

约分:

16

= - \frac{1}{2}

二次方程组的解是:

u = - \frac{5}{8}, u = - \frac{1}{2}

u = cos (x)

代回

cos (x) = - \frac{5}{8}, cos (x) = - \frac{1}{2}

cos (x) = - \frac{5}{8}, cos (x) = - \frac{1}{2}

cos (x) = - \frac{5}{8} : x = arccos (- \frac{5}{8}) + 2 πn, x = - arccos (- \frac{5}{8}) + 2 πn

cos (x) = - \frac{5}{8}

使用反三角函数性质

cos (x) = - \frac{5}{8}

cos (x) = - \frac{5}{8}

的通解

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

x = arccos (- \frac{5}{8}) + 2 πn, x = - arccos (- \frac{5}{8}) + 2 πn

x = arccos (- \frac{5}{8}) + 2 πn, x = - arccos (- \frac{5}{8}) + 2 πn

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

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

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

的通解

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

合并所有解

x = arccos (- \frac{5}{8}) + 2 πn, x = - arccos (- \frac{5}{8}) + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

以小数形式表示解

x = 2.24592 \dots + 2 πn, x = - 2.24592 \dots + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

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