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受欢迎的三角函数 >

3+sin(x)=5cos(2x)

解答

3 + sin (x) = 5 cos (2 x)

解答

x = 0.41151 \dots + 2 πn, x = π - 0.41151 \dots + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

+1

度数

x = 23.57817 \dots^{\circ} + 36 0^{\circ} n, x = 156.42182 \dots^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

求解步骤

3 + sin (x) = 5 cos (2 x)

两边减去

5 cos (2 x)

3 + sin (x) - 5 cos (2 x) = 0

使用三角恒等式改写

3 + sin (x) - 5 cos (2 x)

使用倍角公式:

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

= 3 + sin (x) - 5 (1 - 2 sin^{2} (x))

化简

3 + sin (x) - 5 (1 - 2 sin^{2} (x)) : 10 sin^{2} (x) + sin (x) - 2

3 + sin (x) - 5 (1 - 2 sin^{2} (x))

乘开

- 5 (1 - 2 sin^{2} (x)) : - 5 + 10 sin^{2} (x)

- 5 (1 - 2 sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = - 5, b = 1, c = 2 sin^{2} (x)

= - 5 \cdot 1 - (- 5) \cdot 2 sin^{2} (x)

使用加减运算法则

- (- a) = a

= - 5 \cdot 1 + 5 \cdot 2 sin^{2} (x)

化简

- 5 \cdot 1 + 5 \cdot 2 sin^{2} (x) : - 5 + 10 sin^{2} (x)

- 5 \cdot 1 + 5 \cdot 2 sin^{2} (x)

数字相乘:

5 \cdot 1 = 5

= - 5 + 5 \cdot 2 sin^{2} (x)

数字相乘:

5 \cdot 2 = 10

= - 5 + 10 sin^{2} (x)

= - 5 + 10 sin^{2} (x)

= 3 + sin (x) - 5 + 10 sin^{2} (x)

化简

3 + sin (x) - 5 + 10 sin^{2} (x) : 10 sin^{2} (x) + sin (x) - 2

3 + sin (x) - 5 + 10 sin^{2} (x)

对同类项分组

= sin (x) + 10 sin^{2} (x) + 3 - 5

数字相加/相减:

3 - 5 = - 2

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

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

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

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

用替代法求解

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

令

: sin (x) = u

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

- 2 + u + 10 u^{2} = 0 : u = \frac{2}{5}, u = - \frac{1}{2}

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = 10, b = 1, c = - 2

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

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

1^{2} - 4 \cdot 10 (- 2) = 9

1^{2} - 4 \cdot 10 (- 2)

使用法则

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 10 (- 2)

使用法则

- (- a) = a

= 1 + 4 \cdot 10 \cdot 2

数字相乘:

4 \cdot 10 \cdot 2 = 80

= 1 + 80

数字相加:

1 + 80 = 81

= 81

因式分解数字:

81 = 9^{2}

= 9^{2}

使用根式运算法则:

n a^{n} = a

9^{2} = 9

= 9

u_{1, 2} = \frac{- 1 \pm 9}{2 \cdot 10}

将解分隔开

u_{1} = \frac{- 1 + 9}{2 \cdot 10}, u_{2} = \frac{- 1 - 9}{2 \cdot 10}

u = \frac{- 1 + 9}{2 \cdot 10} : \frac{2}{5}

\frac{- 1 + 9}{2 \cdot 10}

数字相加/相减:

- 1 + 9 = 8

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

数字相乘:

2 \cdot 10 = 20

= \frac{8}{20}

约分:

4

= \frac{2}{5}

u = \frac{- 1 - 9}{2 \cdot 10} : - \frac{1}{2}

\frac{- 1 - 9}{2 \cdot 10}

数字相减:

- 1 - 9 = - 10

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

数字相乘:

2 \cdot 10 = 20

= \frac{- 10}{20}

使用分式法则:

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

= - \frac{10}{20}

约分:

10

= - \frac{1}{2}

二次方程组的解是:

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

u = sin (x)

代回

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

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

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

sin (x) = \frac{2}{5}

使用反三角函数性质

sin (x) = \frac{2}{5}

sin (x) = \frac{2}{5}

的通解

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

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

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

sin (x) = - \frac{1}{2} : x = \frac{7 π}{6} + 2 πn, x = \frac{11 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{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

合并所有解

x = arcsin (\frac{2}{5}) + 2 πn, x = π - arcsin (\frac{2}{5}) + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

以小数形式表示解

x = 0.41151 \dots + 2 πn, x = π - 0.41151 \dots + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

作图

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