zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

6-2sin(θ)-8sin^2(θ)=0

解答

6 - 2 sin (θ) - 8 sin^{2} (θ) = 0

解答

θ = \frac{3 π}{2} + 2 πn, θ = 0.84806 \dots + 2 πn, θ = π - 0.84806 \dots + 2 πn

+1

度数

θ = 27 0^{\circ} + 36 0^{\circ} n, θ = 48.59037 \dots^{\circ} + 36 0^{\circ} n, θ = 131.40962 \dots^{\circ} + 36 0^{\circ} n

求解步骤

6 - 2 sin (θ) - 8 sin^{2} (θ) = 0

用替代法求解

6 - 2 sin (θ) - 8 sin^{2} (θ) = 0

令

: sin (θ) = u

6 - 2 u - 8 u^{2} = 0

6 - 2 u - 8 u^{2} = 0 : u = - 1, u = \frac{3}{4}

6 - 2 u - 8 u^{2} = 0

改写成标准形式

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

- 8 u^{2} - 2 u + 6 = 0

使用求根公式求解

- 8 u^{2} - 2 u + 6 = 0

二次方程求根公式:

若

a = - 8, b = - 2, c = 6

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 ( - 8 ) \cdot 6}{2 ( - 8 )}

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 ( - 8 ) \cdot 6}{2 ( - 8 )}

(- 2)^{2} - 4 (- 8) \cdot 6 = 14

(- 2)^{2} - 4 (- 8) \cdot 6

使用法则

- (- a) = a

= (- 2)^{2} + 4 \cdot 8 \cdot 6

使用指数法则:

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

若

n

是偶数

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

= 2^{2} + 4 \cdot 8 \cdot 6

数字相乘:

4 \cdot 8 \cdot 6 = 192

= 2^{2} + 192

2^{2} = 4

= 4 + 192

数字相加:

4 + 192 = 196

= 196

因式分解数字:

196 = 1 4^{2}

= 1 4^{2}

使用根式运算法则:

n a^{n} = a

1 4^{2} = 14

= 14

u_{1, 2} = \frac{- ( - 2 ) \pm 14}{2 ( - 8 )}

将解分隔开

u_{1} = \frac{- ( - 2 ) + 14}{2 ( - 8 )}, u_{2} = \frac{- ( - 2 ) - 14}{2 ( - 8 )}

u = \frac{- ( - 2 ) + 14}{2 ( - 8 )} : - 1

\frac{- ( - 2 ) + 14}{2 ( - 8 )}

去除括号:

(- a) = - a, - (- a) = a

= \frac{2 + 14}{- 2 \cdot 8}

数字相加:

2 + 14 = 16

= \frac{16}{- 2 \cdot 8}

数字相乘:

2 \cdot 8 = 16

= \frac{16}{- 16}

使用分式法则:

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

= - \frac{16}{16}

使用法则

\frac{a}{a} = 1

= - 1

u = \frac{- ( - 2 ) - 14}{2 ( - 8 )} : \frac{3}{4}

\frac{- ( - 2 ) - 14}{2 ( - 8 )}

去除括号:

(- a) = - a, - (- a) = a

= \frac{2 - 14}{- 2 \cdot 8}

数字相减:

2 - 14 = - 12

= \frac{- 12}{- 2 \cdot 8}

数字相乘:

2 \cdot 8 = 16

= \frac{- 12}{- 16}

使用分式法则:

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

= \frac{12}{16}

约分:

4

= \frac{3}{4}

二次方程组的解是:

u = - 1, u = \frac{3}{4}

u = sin (θ)

代回

sin (θ) = - 1, sin (θ) = \frac{3}{4}

sin (θ) = - 1, sin (θ) = \frac{3}{4}

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

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

sin (θ) = \frac{3}{4}

使用反三角函数性质

sin (θ) = \frac{3}{4}

sin (θ) = \frac{3}{4}

的通解

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

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

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

合并所有解

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

以小数形式表示解

θ = \frac{3 π}{2} + 2 πn, θ = 0.84806 \dots + 2 πn, θ = π - 0.84806 \dots + 2 πn

作图

流行的例子

cos(2θ)=cos(3θ)

cos (2 θ) = cos (3 θ)

solvefor θ,2sin(θ)=1.5

so l v e f or θ, 2 sin (θ) = 1.5

cos (θ) = - 2109

solvefor x,cos(x)(cos(x)-sin(x))=0

so l v e f or x, cos (x) (cos (x) - sin (x)) = 0

sin (x) = 0.72555