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受欢迎的三角函数 >

4cos(2x)+10sin(x)-7=0

解答

4 cos (2 x) + 10 sin (x) - 7 = 0

解答

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn

+1

度数

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 48.59037 \dots^{\circ} + 36 0^{\circ} n, x = 131.40962 \dots^{\circ} + 36 0^{\circ} n

求解步骤

4 cos (2 x) + 10 sin (x) - 7 = 0

使用三角恒等式改写

- 7 + 10 sin (x) + 4 cos (2 x)

使用倍角公式:

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

= - 7 + 10 sin (x) + 4 (1 - 2 sin^{2} (x))

化简

- 7 + 10 sin (x) + 4 (1 - 2 sin^{2} (x)) : 10 sin (x) - 8 sin^{2} (x) - 3

- 7 + 10 sin (x) + 4 (1 - 2 sin^{2} (x))

乘开

4 (1 - 2 sin^{2} (x)) : 4 - 8 sin^{2} (x)

4 (1 - 2 sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 4, b = 1, c = 2 sin^{2} (x)

= 4 \cdot 1 - 4 \cdot 2 sin^{2} (x)

化简

4 \cdot 1 - 4 \cdot 2 sin^{2} (x) : 4 - 8 sin^{2} (x)

4 \cdot 1 - 4 \cdot 2 sin^{2} (x)

数字相乘:

4 \cdot 1 = 4

= 4 - 4 \cdot 2 sin^{2} (x)

数字相乘:

4 \cdot 2 = 8

= 4 - 8 sin^{2} (x)

= 4 - 8 sin^{2} (x)

= - 7 + 10 sin (x) + 4 - 8 sin^{2} (x)

化简

- 7 + 10 sin (x) + 4 - 8 sin^{2} (x) : 10 sin (x) - 8 sin^{2} (x) - 3

- 7 + 10 sin (x) + 4 - 8 sin^{2} (x)

对同类项分组

= 10 sin (x) - 8 sin^{2} (x) - 7 + 4

数字相加/相减:

- 7 + 4 = - 3

= 10 sin (x) - 8 sin^{2} (x) - 3

= 10 sin (x) - 8 sin^{2} (x) - 3

= 10 sin (x) - 8 sin^{2} (x) - 3

- 3 + 10 sin (x) - 8 sin^{2} (x) = 0

用替代法求解

- 3 + 10 sin (x) - 8 sin^{2} (x) = 0

令

: sin (x) = u

- 3 + 10 u - 8 u^{2} = 0

- 3 + 10 u - 8 u^{2} = 0 : u = \frac{1}{2}, u = \frac{3}{4}

- 3 + 10 u - 8 u^{2} = 0

改写成标准形式

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

- 8 u^{2} + 10 u - 3 = 0

使用求根公式求解

- 8 u^{2} + 10 u - 3 = 0

二次方程求根公式:

若

a = - 8, b = 10, c = - 3

u_{1, 2} = \frac{- 10 \pm 1 0 ^{2} - 4 ( - 8 ) ( - 3 )}{2 ( - 8 )}

u_{1, 2} = \frac{- 10 \pm 1 0 ^{2} - 4 ( - 8 ) ( - 3 )}{2 ( - 8 )}

1 0^{2} - 4 (- 8) (- 3) = 2

1 0^{2} - 4 (- 8) (- 3)

使用法则

- (- a) = a

= 1 0^{2} - 4 \cdot 8 \cdot 3

数字相乘:

4 \cdot 8 \cdot 3 = 96

= 1 0^{2} - 96

1 0^{2} = 100

= 100 - 96

数字相减:

100 - 96 = 4

= 4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

u_{1, 2} = \frac{- 10 \pm 2}{2 ( - 8 )}

将解分隔开

u_{1} = \frac{- 10 + 2}{2 ( - 8 )}, u_{2} = \frac{- 10 - 2}{2 ( - 8 )}

u = \frac{- 10 + 2}{2 ( - 8 )} : \frac{1}{2}

\frac{- 10 + 2}{2 ( - 8 )}

去除括号:

(- a) = - a

= \frac{- 10 + 2}{- 2 \cdot 8}

数字相加/相减:

- 10 + 2 = - 8

= \frac{- 8}{- 2 \cdot 8}

数字相乘:

2 \cdot 8 = 16

= \frac{- 8}{- 16}

使用分式法则:

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

= \frac{8}{16}

约分:

8

= \frac{1}{2}

u = \frac{- 10 - 2}{2 ( - 8 )} : \frac{3}{4}

\frac{- 10 - 2}{2 ( - 8 )}

去除括号:

(- a) = - a

= \frac{- 10 - 2}{- 2 \cdot 8}

数字相减:

- 10 - 2 = - 12

= \frac{- 12}{- 2 \cdot 8}

数字相乘:

2 \cdot 8 = 16

= \frac{- 12}{- 16}

使用分式法则:

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

= \frac{12}{16}

约分:

4

= \frac{3}{4}

二次方程组的解是:

u = \frac{1}{2}, u = \frac{3}{4}

u = sin (x)

代回

sin (x) = \frac{1}{2}, sin (x) = \frac{3}{4}

sin (x) = \frac{1}{2}, sin (x) = \frac{3}{4}

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 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}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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

sin (x) = \frac{3}{4}

使用反三角函数性质

sin (x) = \frac{3}{4}

sin (x) = \frac{3}{4}

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

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

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

合并所有解

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = arcsin (\frac{3}{4}) + 2 πn, x = π - arcsin (\frac{3}{4}) + 2 πn

以小数形式表示解

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn

作图

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