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受欢迎的三角函数 >

9sin^2(θ)+19sin(θ)+8=7sin(θ)+4

解答

9 sin^{2} (θ) + 19 sin (θ) + 8 = 7 sin (θ) + 4

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 = 221.81031 \dots^{\circ} + 36 0^{\circ} n

求解步骤

9 sin^{2} (θ) + 19 sin (θ) + 8 = 7 sin (θ) + 4

用替代法求解

9 sin^{2} (θ) + 19 sin (θ) + 8 = 7 sin (θ) + 4

令

: sin (θ) = u

9 u^{2} + 19 u + 8 = 7 u + 4

9 u^{2} + 19 u + 8 = 7 u + 4 : u = - \frac{2}{3}

9 u^{2} + 19 u + 8 = 7 u + 4

将

4

para o lado esquerdo

9 u^{2} + 19 u + 8 = 7 u + 4

两边减去

4

9 u^{2} + 19 u + 8 - 4 = 7 u + 4 - 4

化简

9 u^{2} + 19 u + 4 = 7 u

9 u^{2} + 19 u + 4 = 7 u

将

7 u

para o lado esquerdo

9 u^{2} + 19 u + 4 = 7 u

两边减去

7 u

9 u^{2} + 19 u + 4 - 7 u = 7 u - 7 u

化简

9 u^{2} + 12 u + 4 = 0

9 u^{2} + 12 u + 4 = 0

使用求根公式求解

9 u^{2} + 12 u + 4 = 0

二次方程求根公式:

若

a = 9, b = 12, c = 4

u_{1, 2} = \frac{- 12 \pm 1 2 ^{2} - 4 \cdot 9 \cdot 4}{2 \cdot 9}

u_{1, 2} = \frac{- 12 \pm 1 2 ^{2} - 4 \cdot 9 \cdot 4}{2 \cdot 9}

1 2^{2} - 4 \cdot 9 \cdot 4 = 0

1 2^{2} - 4 \cdot 9 \cdot 4

数字相乘:

4 \cdot 9 \cdot 4 = 144

= 1 2^{2} - 144

1 2^{2} = 144

= 144 - 144

数字相减:

144 - 144 = 0

= 0

u_{1, 2} = \frac{- 12 \pm 0}{2 \cdot 9}

u = \frac{- 12}{2 \cdot 9}

\frac{- 12}{2 \cdot 9} = - \frac{2}{3}

\frac{- 12}{2 \cdot 9}

数字相乘:

2 \cdot 9 = 18

= \frac{- 12}{18}

使用分式法则:

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

= - \frac{12}{18}

约分:

6

= - \frac{2}{3}

u = - \frac{2}{3}

二次方程组的解是:

u = - \frac{2}{3}

u = sin (θ)

代回

sin (θ) = - \frac{2}{3}

sin (θ) = - \frac{2}{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

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

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