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受欢迎的三角函数 >

-9cos^2(θ)-9sin(θ)+10=-1

解答

- 9 cos^{2} (θ) - 9 sin (θ) + 10 = - 1

解答

θ = 0.72972 \dots + 2 πn, θ = π - 0.72972 \dots + 2 πn, θ = 0.33983 \dots + 2 πn, θ = π - 0.33983 \dots + 2 πn

+1

度数

θ = 41.81031 \dots^{\circ} + 36 0^{\circ} n, θ = 138.18968 \dots^{\circ} + 36 0^{\circ} n, θ = 19.47122 \dots^{\circ} + 36 0^{\circ} n, θ = 160.52877 \dots^{\circ} + 36 0^{\circ} n

求解步骤

- 9 cos^{2} (θ) - 9 sin (θ) + 10 = - 1

两边减去

- 1

- 9 cos^{2} (θ) - 9 sin (θ) + 11 = 0

使用三角恒等式改写

11 - 9 cos^{2} (θ) - 9 sin (θ)

使用毕达哥拉斯恒等式:

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

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

= 11 - 9 (1 - sin^{2} (θ)) - 9 sin (θ)

化简

11 - 9 (1 - sin^{2} (θ)) - 9 sin (θ) : 9 sin^{2} (θ) - 9 sin (θ) + 2

11 - 9 (1 - sin^{2} (θ)) - 9 sin (θ)

乘开

- 9 (1 - sin^{2} (θ)) : - 9 + 9 sin^{2} (θ)

- 9 (1 - sin^{2} (θ))

使用分配律:

a (b - c) = ab - a c

a = - 9, b = 1, c = sin^{2} (θ)

= - 9 \cdot 1 - (- 9) sin^{2} (θ)

使用加减运算法则

- (- a) = a

= - 9 \cdot 1 + 9 sin^{2} (θ)

数字相乘:

9 \cdot 1 = 9

= - 9 + 9 sin^{2} (θ)

= 11 - 9 + 9 sin^{2} (θ) - 9 sin (θ)

数字相减:

11 - 9 = 2

= 9 sin^{2} (θ) - 9 sin (θ) + 2

= 9 sin^{2} (θ) - 9 sin (θ) + 2

2 - 9 sin (θ) + 9 sin^{2} (θ) = 0

用替代法求解

2 - 9 sin (θ) + 9 sin^{2} (θ) = 0

令

: sin (θ) = u

2 - 9 u + 9 u^{2} = 0

2 - 9 u + 9 u^{2} = 0 : u = \frac{2}{3}, u = \frac{1}{3}

2 - 9 u + 9 u^{2} = 0

改写成标准形式

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

9 u^{2} - 9 u + 2 = 0

使用求根公式求解

9 u^{2} - 9 u + 2 = 0

二次方程求根公式:

若

a = 9, b = - 9, c = 2

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

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

(- 9)^{2} - 4 \cdot 9 \cdot 2 = 3

(- 9)^{2} - 4 \cdot 9 \cdot 2

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- 9)^{2} = 9^{2}

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

数字相乘:

4 \cdot 9 \cdot 2 = 72

= 9^{2} - 72

9^{2} = 81

= 81 - 72

数字相减:

81 - 72 = 9

= 9

因式分解数字:

9 = 3^{2}

= 3^{2}

使用根式运算法则:

n a^{n} = a

3^{2} = 3

= 3

u_{1, 2} = \frac{- ( - 9 ) \pm 3}{2 \cdot 9}

将解分隔开

u_{1} = \frac{- ( - 9 ) + 3}{2 \cdot 9}, u_{2} = \frac{- ( - 9 ) - 3}{2 \cdot 9}

u = \frac{- ( - 9 ) + 3}{2 \cdot 9} : \frac{2}{3}

\frac{- ( - 9 ) + 3}{2 \cdot 9}

使用法则

- (- a) = a

= \frac{9 + 3}{2 \cdot 9}

数字相加:

9 + 3 = 12

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

数字相乘:

2 \cdot 9 = 18

= \frac{12}{18}

约分:

6

= \frac{2}{3}

u = \frac{- ( - 9 ) - 3}{2 \cdot 9} : \frac{1}{3}

\frac{- ( - 9 ) - 3}{2 \cdot 9}

使用法则

- (- a) = a

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

数字相减:

9 - 3 = 6

= \frac{6}{2 \cdot 9}

数字相乘:

2 \cdot 9 = 18

= \frac{6}{18}

约分:

6

= \frac{1}{3}

二次方程组的解是:

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

u = sin (θ)

代回

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

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

sin (θ) = \frac{1}{3}

使用反三角函数性质

sin (θ) = \frac{1}{3}

sin (θ) = \frac{1}{3}

的通解

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

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

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

合并所有解

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

以小数形式表示解

θ = 0.72972 \dots + 2 πn, θ = π - 0.72972 \dots + 2 πn, θ = 0.33983 \dots + 2 πn, θ = π - 0.33983 \dots + 2 πn

作图

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