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受欢迎的三角函数 >

4sin^2(x)+7sin(x)=2

解答

4 sin^{2} (x) + 7 sin (x) = 2

解答

x = 0.25268 \dots + 2 πn, x = π - 0.25268 \dots + 2 πn

+1

度数

x = 14.47751 \dots^{\circ} + 36 0^{\circ} n, x = 165.52248 \dots^{\circ} + 36 0^{\circ} n

求解步骤

4 sin^{2} (x) + 7 sin (x) = 2

用替代法求解

4 sin^{2} (x) + 7 sin (x) = 2

令

: sin (x) = u

4 u^{2} + 7 u = 2

4 u^{2} + 7 u = 2 : u = \frac{1}{4}, u = - 2

4 u^{2} + 7 u = 2

将

2

para o lado esquerdo

4 u^{2} + 7 u = 2

两边减去

2

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

化简

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = 4, b = 7, c = - 2

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

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

7^{2} - 4 \cdot 4 (- 2) = 9

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

使用法则

- (- a) = a

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

数字相乘:

4 \cdot 4 \cdot 2 = 32

= 7^{2} + 32

7^{2} = 49

= 49 + 32

数字相加:

49 + 32 = 81

= 81

因式分解数字:

81 = 9^{2}

= 9^{2}

使用根式运算法则:

n a^{n} = a

9^{2} = 9

= 9

u_{1, 2} = \frac{- 7 \pm 9}{2 \cdot 4}

将解分隔开

u_{1} = \frac{- 7 + 9}{2 \cdot 4}, u_{2} = \frac{- 7 - 9}{2 \cdot 4}

u = \frac{- 7 + 9}{2 \cdot 4} : \frac{1}{4}

\frac{- 7 + 9}{2 \cdot 4}

数字相加/相减:

- 7 + 9 = 2

= \frac{2}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{2}{8}

约分:

2

= \frac{1}{4}

u = \frac{- 7 - 9}{2 \cdot 4} : - 2

\frac{- 7 - 9}{2 \cdot 4}

数字相减:

- 7 - 9 = - 16

= \frac{- 16}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- 16}{8}

使用分式法则:

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

= - \frac{16}{8}

数字相除:

\frac{16}{8} = 2

= - 2

二次方程组的解是:

u = \frac{1}{4}, u = - 2

u = sin (x)

代回

sin (x) = \frac{1}{4}, sin (x) = - 2

sin (x) = \frac{1}{4}, sin (x) = - 2

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

sin (x) = \frac{1}{4}

使用反三角函数性质

sin (x) = \frac{1}{4}

sin (x) = \frac{1}{4}

的通解

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

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

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

sin (x) = - 2 :

无解

sin (x) = - 2

- 1 \leq sin (x) \leq 1

无解

合并所有解

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

以小数形式表示解

x = 0.25268 \dots + 2 πn, x = π - 0.25268 \dots + 2 πn

作图

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