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 >

-7sin^2(θ)+2sin(θ)+7=0

解答

- 7 sin^{2} (θ) + 2 sin (θ) + 7 = 0

解答

θ = - 1.04974 \dots + 2 πn, θ = π + 1.04974 \dots + 2 πn

+1

度数

θ = - 60.14585 \dots^{\circ} + 36 0^{\circ} n, θ = 240.14585 \dots^{\circ} + 36 0^{\circ} n

求解步骤

- 7 sin^{2} (θ) + 2 sin (θ) + 7 = 0

用替代法求解

- 7 sin^{2} (θ) + 2 sin (θ) + 7 = 0

令

: sin (θ) = u

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

- 7 u^{2} + 2 u + 7 = 0 : u = - \frac{- 1 + 5 2}{7}, u = \frac{1 + 5 2}{7}

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

使用求根公式求解

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

二次方程求根公式:

若

a = - 7, b = 2, c = 7

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

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

2^{2} - 4 (- 7) \cdot 7 = 102

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

使用法则

- (- a) = a

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

数字相乘:

4 \cdot 7 \cdot 7 = 196

= 2^{2} + 196

2^{2} = 4

= 4 + 196

数字相加:

4 + 196 = 200

= 200

200

质因数分解:

2^{3} \cdot 5^{2}

200

200

除以

2200 = 100 \cdot 2

= 2 \cdot 100

100

除以

2100 = 50 \cdot 2

= 2 \cdot 2 \cdot 50

50

除以

250 = 25 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 25

25

除以

525 = 5 \cdot 5

= 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5

2, 5

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5

= 2^{3} \cdot 5^{2}

= 2^{3} \cdot 5^{2}

使用指数法则:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 5^{2} \cdot 2

使用根式运算法则:

n ab = n a n b

= 2 2^{2} 5^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 22 5^{2}

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 2 \cdot 52

整理后得

= 102

u_{1, 2} = \frac{- 2 \pm 10 2}{2 ( - 7 )}

将解分隔开

u_{1} = \frac{- 2 + 10 2}{2 ( - 7 )}, u_{2} = \frac{- 2 - 10 2}{2 ( - 7 )}

u = \frac{- 2 + 10 2}{2 ( - 7 )} : - \frac{- 1 + 5 2}{7}

\frac{- 2 + 10 2}{2 ( - 7 )}

去除括号:

(- a) = - a

= \frac{- 2 + 10 2}{- 2 \cdot 7}

数字相乘:

2 \cdot 7 = 14

= \frac{- 2 + 10 2}{- 14}

使用分式法则:

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

= - \frac{- 2 + 10 2}{14}

消掉

\frac{- 2 + 10 2}{14} : \frac{5 2 - 1}{7}

\frac{- 2 + 10 2}{14}

分解

- 2 + 102 : 2 (- 1 + 52)

- 2 + 102

改写为

= - 2 \cdot 1 + 2 \cdot 52

因式分解出通项

2

= 2 (- 1 + 52)

= \frac{2 ( - 1 + 5 2 )}{14}

约分:

2

= \frac{- 1 + 5 2}{7}

= - \frac{5 2 - 1}{7}

= - \frac{- 1 + 5 2}{7}

u = \frac{- 2 - 10 2}{2 ( - 7 )} : \frac{1 + 5 2}{7}

\frac{- 2 - 10 2}{2 ( - 7 )}

去除括号:

(- a) = - a

= \frac{- 2 - 10 2}{- 2 \cdot 7}

数字相乘:

2 \cdot 7 = 14

= \frac{- 2 - 10 2}{- 14}

使用分式法则:

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

- 2 - 102 = - (2 + 102)

= \frac{2 + 10 2}{14}

分解

2 + 102 : 2 (1 + 52)

2 + 102

改写为

= 2 \cdot 1 + 2 \cdot 52

因式分解出通项

2

= 2 (1 + 52)

= \frac{2 ( 1 + 5 2 )}{14}

约分:

2

= \frac{1 + 5 2}{7}

二次方程组的解是:

u = - \frac{- 1 + 5 2}{7}, u = \frac{1 + 5 2}{7}

u = sin (θ)

代回

sin (θ) = - \frac{- 1 + 5 2}{7}, sin (θ) = \frac{1 + 5 2}{7}

sin (θ) = - \frac{- 1 + 5 2}{7}, sin (θ) = \frac{1 + 5 2}{7}

sin (θ) = - \frac{- 1 + 5 2}{7} : θ = arcsin (- \frac{- 1 + 5 2}{7}) + 2 πn, θ = π + arcsin (\frac{- 1 + 5 2}{7}) + 2 πn

sin (θ) = - \frac{- 1 + 5 2}{7}

使用反三角函数性质

sin (θ) = - \frac{- 1 + 5 2}{7}

sin (θ) = - \frac{- 1 + 5 2}{7}

的通解

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

θ = arcsin (- \frac{- 1 + 5 2}{7}) + 2 πn, θ = π + arcsin (\frac{- 1 + 5 2}{7}) + 2 πn

θ = arcsin (- \frac{- 1 + 5 2}{7}) + 2 πn, θ = π + arcsin (\frac{- 1 + 5 2}{7}) + 2 πn

sin (θ) = \frac{1 + 5 2}{7} :

无解

sin (θ) = \frac{1 + 5 2}{7}

- 1 \leq sin (x) \leq 1

无解

合并所有解

θ = arcsin (- \frac{- 1 + 5 2}{7}) + 2 πn, θ = π + arcsin (\frac{- 1 + 5 2}{7}) + 2 πn

以小数形式表示解

θ = - 1.04974 \dots + 2 πn, θ = π + 1.04974 \dots + 2 πn

作图

流行的例子

cos(x)-2sin^2(x)+1=0

cos (x) - 2 sin^{2} (x) + 1 = 0

3tan(2x)+sqrt(3)=0,0<= x<2pi

3 tan (2 x) + 3 = 0, 0 \leq x < 2 π

sin(2x)-sin(x)cos(x)=cos(x)

sin (2 x) - sin (x) cos (x) = cos (x)

cot^2(x)=1-csc(x)

cot^{2} (x) = 1 - csc (x)

2sin^2(θ)+9sin(θ)-5=0

2 sin^{2} (θ) + 9 sin (θ) - 5 = 0