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受欢迎的三角函数 >

-2cos(2θ)+3sin(θ)+4=3

解答

- 2 cos (2 θ) + 3 sin (θ) + 4 = 3

解答

θ = 0.25268 \dots + 2 πn, θ = π - 0.25268 \dots + 2 πn, θ = \frac{3 π}{2} + 2 πn

+1

度数

θ = 14.47751 \dots^{\circ} + 36 0^{\circ} n, θ = 165.52248 \dots^{\circ} + 36 0^{\circ} n, θ = 27 0^{\circ} + 36 0^{\circ} n

求解步骤

- 2 cos (2 θ) + 3 sin (θ) + 4 = 3

两边减去

3

3 sin (θ) - 2 cos (2 θ) + 1 = 0

使用三角恒等式改写

1 - 2 cos (2 θ) + 3 sin (θ)

使用倍角公式:

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

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

化简

1 - 2 (1 - 2 sin^{2} (θ)) + 3 sin (θ) : 4 sin^{2} (θ) + 3 sin (θ) - 1

1 - 2 (1 - 2 sin^{2} (θ)) + 3 sin (θ)

乘开

- 2 (1 - 2 sin^{2} (θ)) : - 2 + 4 sin^{2} (θ)

- 2 (1 - 2 sin^{2} (θ))

使用分配律:

a (b - c) = ab - a c

a = - 2, b = 1, c = 2 sin^{2} (θ)

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

使用加减运算法则

- (- a) = a

= - 2 \cdot 1 + 2 \cdot 2 sin^{2} (θ)

化简

- 2 \cdot 1 + 2 \cdot 2 sin^{2} (θ) : - 2 + 4 sin^{2} (θ)

- 2 \cdot 1 + 2 \cdot 2 sin^{2} (θ)

数字相乘:

2 \cdot 1 = 2

= - 2 + 2 \cdot 2 sin^{2} (θ)

数字相乘:

2 \cdot 2 = 4

= - 2 + 4 sin^{2} (θ)

= - 2 + 4 sin^{2} (θ)

= 1 - 2 + 4 sin^{2} (θ) + 3 sin (θ)

数字相减:

1 - 2 = - 1

= 4 sin^{2} (θ) + 3 sin (θ) - 1

= 4 sin^{2} (θ) + 3 sin (θ) - 1

- 1 + 3 sin (θ) + 4 sin^{2} (θ) = 0

用替代法求解

- 1 + 3 sin (θ) + 4 sin^{2} (θ) = 0

令

: sin (θ) = u

- 1 + 3 u + 4 u^{2} = 0

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

- 1 + 3 u + 4 u^{2} = 0

改写成标准形式

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

4 u^{2} + 3 u - 1 = 0

使用求根公式求解

4 u^{2} + 3 u - 1 = 0

二次方程求根公式:

若

a = 4, b = 3, c = - 1

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

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

3^{2} - 4 \cdot 4 (- 1) = 5

3^{2} - 4 \cdot 4 (- 1)

使用法则

- (- a) = a

= 3^{2} + 4 \cdot 4 \cdot 1

数字相乘:

4 \cdot 4 \cdot 1 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

数字相加:

9 + 16 = 25

= 25

因式分解数字:

25 = 5^{2}

= 5^{2}

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- 3 \pm 5}{2 \cdot 4}

将解分隔开

u_{1} = \frac{- 3 + 5}{2 \cdot 4}, u_{2} = \frac{- 3 - 5}{2 \cdot 4}

u = \frac{- 3 + 5}{2 \cdot 4} : \frac{1}{4}

\frac{- 3 + 5}{2 \cdot 4}

数字相加/相减:

- 3 + 5 = 2

= \frac{2}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{2}{8}

约分:

2

= \frac{1}{4}

u = \frac{- 3 - 5}{2 \cdot 4} : - 1

\frac{- 3 - 5}{2 \cdot 4}

数字相减:

- 3 - 5 = - 8

= \frac{- 8}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- 8}{8}

使用分式法则:

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

= - \frac{8}{8}

使用法则

\frac{a}{a} = 1

= - 1

二次方程组的解是:

u = \frac{1}{4}, u = - 1

u = sin (θ)

代回

sin (θ) = \frac{1}{4}, sin (θ) = - 1

sin (θ) = \frac{1}{4}, sin (θ) = - 1

sin (θ) = \frac{1}{4} : θ = arcsin (\frac{1}{4}) + 2 πn, θ = π - arcsin (\frac{1}{4}) + 2 πn

sin (θ) = \frac{1}{4}

使用反三角函数性质

sin (θ) = \frac{1}{4}

sin (θ) = \frac{1}{4}

的通解

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

θ = arcsin (\frac{1}{4}) + 2 πn, θ = π - arcsin (\frac{1}{4}) + 2 πn

θ = arcsin (\frac{1}{4}) + 2 πn, θ = π - arcsin (\frac{1}{4}) + 2 πn

sin (θ) = - 1 : θ = \frac{3 π}{2} + 2 πn

sin (θ) = - 1

sin (θ) = - 1

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}

θ = \frac{3 π}{2} + 2 πn

θ = \frac{3 π}{2} + 2 πn

合并所有解

θ = arcsin (\frac{1}{4}) + 2 πn, θ = π - arcsin (\frac{1}{4}) + 2 πn, θ = \frac{3 π}{2} + 2 πn

以小数形式表示解

θ = 0.25268 \dots + 2 πn, θ = π - 0.25268 \dots + 2 πn, θ = \frac{3 π}{2} + 2 πn

作图

流行的例子

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sin(θ)+cos(θ)=-sqrt(2)

sin (θ) + cos (θ) = - 2

cot (θ) = 3.2404

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