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 >

4sin^2(x)+2sin(x)-1=0

解答

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

解答

x = 0.31415 \dots + 2 πn, x = π - 0.31415 \dots + 2 πn, x = - 0.94247 \dots + 2 πn, x = π + 0.94247 \dots + 2 πn

+1

度数

x = 1 8^{\circ} + 36 0^{\circ} n, x = 16 2^{\circ} + 36 0^{\circ} n, x = - 5 4^{\circ} + 36 0^{\circ} n, x = 23 4^{\circ} + 36 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: sin (x) = u

4 u^{2} + 2 u - 1 = 0

4 u^{2} + 2 u - 1 = 0 : u = \frac{- 1 + 5}{4}, u = - \frac{1 + 5}{4}

4 u^{2} + 2 u - 1 = 0

使用求根公式求解

4 u^{2} + 2 u - 1 = 0

二次方程求根公式:

若

a = 4, b = 2, c = - 1

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

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

2^{2} - 4 \cdot 4 (- 1) = 25

2^{2} - 4 \cdot 4 (- 1)

使用法则

- (- a) = a

= 2^{2} + 4 \cdot 4 \cdot 1

数字相乘:

4 \cdot 4 \cdot 1 = 16

= 2^{2} + 16

2^{2} = 4

= 4 + 16

数字相加:

4 + 16 = 20

= 20

20

质因数分解:

2^{2} \cdot 5

20

20

除以

220 = 10 \cdot 2

= 2 \cdot 10

10

除以

210 = 5 \cdot 2

= 2 \cdot 2 \cdot 5

2, 5

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 5

= 2^{2} \cdot 5

= 2^{2} \cdot 5

使用根式运算法则:

n ab = n a n b

= 5 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 25

u_{1, 2} = \frac{- 2 \pm 2 5}{2 \cdot 4}

将解分隔开

u_{1} = \frac{- 2 + 2 5}{2 \cdot 4}, u_{2} = \frac{- 2 - 2 5}{2 \cdot 4}

u = \frac{- 2 + 2 5}{2 \cdot 4} : \frac{- 1 + 5}{4}

\frac{- 2 + 2 5}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- 2 + 2 5}{8}

分解

- 2 + 25 : 2 (- 1 + 5)

- 2 + 25

改写为

= - 2 \cdot 1 + 25

因式分解出通项

2

= 2 (- 1 + 5)

= \frac{2 ( - 1 + 5 )}{8}

约分:

2

= \frac{- 1 + 5}{4}

u = \frac{- 2 - 2 5}{2 \cdot 4} : - \frac{1 + 5}{4}

\frac{- 2 - 2 5}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- 2 - 2 5}{8}

分解

- 2 - 25 : - 2 (1 + 5)

- 2 - 25

改写为

= - 2 \cdot 1 - 25

因式分解出通项

2

= - 2 (1 + 5)

= - \frac{2 ( 1 + 5 )}{8}

约分:

2

= - \frac{1 + 5}{4}

二次方程组的解是:

u = \frac{- 1 + 5}{4}, u = - \frac{1 + 5}{4}

u = sin (x)

代回

sin (x) = \frac{- 1 + 5}{4}, sin (x) = - \frac{1 + 5}{4}

sin (x) = \frac{- 1 + 5}{4}, sin (x) = - \frac{1 + 5}{4}

sin (x) = \frac{- 1 + 5}{4} : x = arcsin (\frac{- 1 + 5}{4}) + 2 πn, x = π - arcsin (\frac{- 1 + 5}{4}) + 2 πn

sin (x) = \frac{- 1 + 5}{4}

使用反三角函数性质

sin (x) = \frac{- 1 + 5}{4}

sin (x) = \frac{- 1 + 5}{4}

的通解

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

x = arcsin (\frac{- 1 + 5}{4}) + 2 πn, x = π - arcsin (\frac{- 1 + 5}{4}) + 2 πn

x = arcsin (\frac{- 1 + 5}{4}) + 2 πn, x = π - arcsin (\frac{- 1 + 5}{4}) + 2 πn

sin (x) = - \frac{1 + 5}{4} : x = arcsin (- \frac{1 + 5}{4}) + 2 πn, x = π + arcsin (\frac{1 + 5}{4}) + 2 πn

sin (x) = - \frac{1 + 5}{4}

使用反三角函数性质

sin (x) = - \frac{1 + 5}{4}

sin (x) = - \frac{1 + 5}{4}

的通解

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

x = arcsin (- \frac{1 + 5}{4}) + 2 πn, x = π + arcsin (\frac{1 + 5}{4}) + 2 πn

x = arcsin (- \frac{1 + 5}{4}) + 2 πn, x = π + arcsin (\frac{1 + 5}{4}) + 2 πn

合并所有解

x = arcsin (\frac{- 1 + 5}{4}) + 2 πn, x = π - arcsin (\frac{- 1 + 5}{4}) + 2 πn, x = arcsin (- \frac{1 + 5}{4}) + 2 πn, x = π + arcsin (\frac{1 + 5}{4}) + 2 πn

以小数形式表示解

x = 0.31415 \dots + 2 πn, x = π - 0.31415 \dots + 2 πn, x = - 0.94247 \dots + 2 πn, x = π + 0.94247 \dots + 2 πn

作图

流行的例子

tan (θ) = \frac{1}{6}

cos (x) = - 12

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

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

sin (x - 3 0^{\circ}) = 0

sin (A) = 0.4