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受欢迎的三角函数 >

3sin^2(θ)+3sin(θ)-6=-4sin(θ)

解答

3 sin^{2} (θ) + 3 sin (θ) - 6 = - 4 sin (θ)

解答

θ = 0.72972 \dots + 2 πn, θ = π - 0.72972 \dots + 2 πn

+1

度数

θ = 41.81031 \dots^{\circ} + 36 0^{\circ} n, θ = 138.18968 \dots^{\circ} + 36 0^{\circ} n

求解步骤

3 sin^{2} (θ) + 3 sin (θ) - 6 = - 4 sin (θ)

用替代法求解

3 sin^{2} (θ) + 3 sin (θ) - 6 = - 4 sin (θ)

令

: sin (θ) = u

3 u^{2} + 3 u - 6 = - 4 u

3 u^{2} + 3 u - 6 = - 4 u : u = \frac{2}{3}, u = - 3

3 u^{2} + 3 u - 6 = - 4 u

将

4 u

para o lado esquerdo

3 u^{2} + 3 u - 6 = - 4 u

两边加上

4 u

3 u^{2} + 3 u - 6 + 4 u = - 4 u + 4 u

化简

3 u^{2} + 7 u - 6 = 0

3 u^{2} + 7 u - 6 = 0

使用求根公式求解

3 u^{2} + 7 u - 6 = 0

二次方程求根公式:

若

a = 3, b = 7, c = - 6

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 \cdot 3 ( - 6 )}{2 \cdot 3}

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 \cdot 3 ( - 6 )}{2 \cdot 3}

7^{2} - 4 \cdot 3 (- 6) = 11

7^{2} - 4 \cdot 3 (- 6)

使用法则

- (- a) = a

= 7^{2} + 4 \cdot 3 \cdot 6

数字相乘:

4 \cdot 3 \cdot 6 = 72

= 7^{2} + 72

7^{2} = 49

= 49 + 72

数字相加:

49 + 72 = 121

= 121

因式分解数字:

121 = 1 1^{2}

= 1 1^{2}

使用根式运算法则:

n a^{n} = a

1 1^{2} = 11

= 11

u_{1, 2} = \frac{- 7 \pm 11}{2 \cdot 3}

将解分隔开

u_{1} = \frac{- 7 + 11}{2 \cdot 3}, u_{2} = \frac{- 7 - 11}{2 \cdot 3}

u = \frac{- 7 + 11}{2 \cdot 3} : \frac{2}{3}

\frac{- 7 + 11}{2 \cdot 3}

数字相加/相减:

- 7 + 11 = 4

= \frac{4}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{4}{6}

约分:

2

= \frac{2}{3}

u = \frac{- 7 - 11}{2 \cdot 3} : - 3

\frac{- 7 - 11}{2 \cdot 3}

数字相减:

- 7 - 11 = - 18

= \frac{- 18}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{- 18}{6}

使用分式法则:

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

= - \frac{18}{6}

数字相除:

\frac{18}{6} = 3

= - 3

二次方程组的解是:

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

u = sin (θ)

代回

sin (θ) = \frac{2}{3}, sin (θ) = - 3

sin (θ) = \frac{2}{3}, sin (θ) = - 3

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

sin (θ) = \frac{2}{3}

使用反三角函数性质

sin (θ) = \frac{2}{3}

sin (θ) = \frac{2}{3}

的通解

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

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

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

sin (θ) = - 3 :

无解

sin (θ) = - 3

- 1 \leq sin (x) \leq 1

无解

合并所有解

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

以小数形式表示解

θ = 0.72972 \dots + 2 πn, θ = π - 0.72972 \dots + 2 πn

作图

流行的例子

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tan (x) = \frac{80}{50}

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