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受欢迎的三角函数 >

3sin^2(x)=5sin(x)+2

解答

3 sin^{2} (x) = 5 sin (x) + 2

解答

x = - 0.33983 \dots + 2 πn, x = π + 0.33983 \dots + 2 πn

+1

度数

x = - 19.47122 \dots^{\circ} + 36 0^{\circ} n, x = 199.47122 \dots^{\circ} + 36 0^{\circ} n

求解步骤

3 sin^{2} (x) = 5 sin (x) + 2

用替代法求解

3 sin^{2} (x) = 5 sin (x) + 2

令

: sin (x) = u

3 u^{2} = 5 u + 2

3 u^{2} = 5 u + 2 : u = 2, u = - \frac{1}{3}

3 u^{2} = 5 u + 2

将

2

para o lado esquerdo

3 u^{2} = 5 u + 2

两边减去

2

3 u^{2} - 2 = 5 u + 2 - 2

化简

3 u^{2} - 2 = 5 u

3 u^{2} - 2 = 5 u

将

5 u

para o lado esquerdo

3 u^{2} - 2 = 5 u

两边减去

5 u

3 u^{2} - 2 - 5 u = 5 u - 5 u

化简

3 u^{2} - 2 - 5 u = 0

3 u^{2} - 2 - 5 u = 0

改写成标准形式

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

3 u^{2} - 5 u - 2 = 0

使用求根公式求解

3 u^{2} - 5 u - 2 = 0

二次方程求根公式:

若

a = 3, b = - 5, c = - 2

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 3 ( - 2 )}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 3 ( - 2 )}{2 \cdot 3}

(- 5)^{2} - 4 \cdot 3 (- 2) = 7

(- 5)^{2} - 4 \cdot 3 (- 2)

使用法则

- (- a) = a

= (- 5)^{2} + 4 \cdot 3 \cdot 2

使用指数法则:

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

若

n

是偶数

(- 5)^{2} = 5^{2}

= 5^{2} + 4 \cdot 3 \cdot 2

数字相乘:

4 \cdot 3 \cdot 2 = 24

= 5^{2} + 24

5^{2} = 25

= 25 + 24

数字相加:

25 + 24 = 49

= 49

因式分解数字:

49 = 7^{2}

= 7^{2}

使用根式运算法则:

n a^{n} = a

7^{2} = 7

= 7

u_{1, 2} = \frac{- ( - 5 ) \pm 7}{2 \cdot 3}

将解分隔开

u_{1} = \frac{- ( - 5 ) + 7}{2 \cdot 3}, u_{2} = \frac{- ( - 5 ) - 7}{2 \cdot 3}

u = \frac{- ( - 5 ) + 7}{2 \cdot 3} : 2

\frac{- ( - 5 ) + 7}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{5 + 7}{2 \cdot 3}

数字相加:

5 + 7 = 12

= \frac{12}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{12}{6}

数字相除:

\frac{12}{6} = 2

= 2

u = \frac{- ( - 5 ) - 7}{2 \cdot 3} : - \frac{1}{3}

\frac{- ( - 5 ) - 7}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{5 - 7}{2 \cdot 3}

数字相减:

5 - 7 = - 2

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

数字相乘:

2 \cdot 3 = 6

= \frac{- 2}{6}

使用分式法则:

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

= - \frac{2}{6}

约分:

2

= - \frac{1}{3}

二次方程组的解是:

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

u = sin (x)

代回

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

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

sin (x) = 2 :

无解

sin (x) = 2

- 1 \leq sin (x) \leq 1

无解

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

sin (x) = - \frac{1}{3}

使用反三角函数性质

sin (x) = - \frac{1}{3}

sin (x) = - \frac{1}{3}

的通解

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

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

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

合并所有解

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

以小数形式表示解

x = - 0.33983 \dots + 2 πn, x = π + 0.33983 \dots + 2 πn

作图

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