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受欢迎的三角函数 >

sin^2(x)+6sin(x)=5

解答

sin^{2} (x) + 6 sin (x) = 5

解答

x = 0.83553 \dots + 2 πn, x = π - 0.83553 \dots + 2 πn

+1

度数

x = 47.87279 \dots^{\circ} + 36 0^{\circ} n, x = 132.12720 \dots^{\circ} + 36 0^{\circ} n

求解步骤

sin^{2} (x) + 6 sin (x) = 5

用替代法求解

sin^{2} (x) + 6 sin (x) = 5

令

: sin (x) = u

u^{2} + 6 u = 5

u^{2} + 6 u = 5 : u = - 3 + 14, u = - 3 - 14

u^{2} + 6 u = 5

将

5

para o lado esquerdo

u^{2} + 6 u = 5

两边减去

5

u^{2} + 6 u - 5 = 5 - 5

化简

u^{2} + 6 u - 5 = 0

u^{2} + 6 u - 5 = 0

使用求根公式求解

u^{2} + 6 u - 5 = 0

二次方程求根公式:

若

a = 1, b = 6, c = - 5

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 1 \cdot ( - 5 )}{2 \cdot 1}

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 1 \cdot ( - 5 )}{2 \cdot 1}

6^{2} - 4 \cdot 1 \cdot (- 5) = 214

6^{2} - 4 \cdot 1 \cdot (- 5)

使用法则

- (- a) = a

= 6^{2} + 4 \cdot 1 \cdot 5

数字相乘:

4 \cdot 1 \cdot 5 = 20

= 6^{2} + 20

6^{2} = 36

= 36 + 20

数字相加:

36 + 20 = 56

= 56

56

质因数分解:

2^{3} \cdot 7

56

56

除以

256 = 28 \cdot 2

= 2 \cdot 28

28

除以

228 = 14 \cdot 2

= 2 \cdot 2 \cdot 14

14

除以

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 7

2, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2 \cdot 7

= 2^{3} \cdot 7

= 2^{3} \cdot 7

使用指数法则:

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

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

使用根式运算法则:

n ab = n a n b

= 2^{2} 2 \cdot 7

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2 2 \cdot 7

整理后得

= 214

u_{1, 2} = \frac{- 6 \pm 2 14}{2 \cdot 1}

将解分隔开

u_{1} = \frac{- 6 + 2 14}{2 \cdot 1}, u_{2} = \frac{- 6 - 2 14}{2 \cdot 1}

u = \frac{- 6 + 2 14}{2 \cdot 1} : - 3 + 14

\frac{- 6 + 2 14}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 6 + 2 14}{2}

分解

- 6 + 214 : 2 (- 3 + 14)

- 6 + 214

改写为

= - 2 \cdot 3 + 214

因式分解出通项

2

= 2 (- 3 + 14)

= \frac{2 ( - 3 + 14 )}{2}

数字相除:

\frac{2}{2} = 1

= - 3 + 14

u = \frac{- 6 - 2 14}{2 \cdot 1} : - 3 - 14

\frac{- 6 - 2 14}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 6 - 2 14}{2}

分解

- 6 - 214 : - 2 (3 + 14)

- 6 - 214

改写为

= - 2 \cdot 3 - 214

因式分解出通项

2

= - 2 (3 + 14)

= - \frac{2 ( 3 + 14 )}{2}

数字相除:

\frac{2}{2} = 1

= - (3 + 14)

取负

- (3 + 14) = - 3 - 14

= - 3 - 14

二次方程组的解是:

u = - 3 + 14, u = - 3 - 14

u = sin (x)

代回

sin (x) = - 3 + 14, sin (x) = - 3 - 14

sin (x) = - 3 + 14, sin (x) = - 3 - 14

sin (x) = - 3 + 14 : x = arcsin (- 3 + 14) + 2 πn, x = π - arcsin (- 3 + 14) + 2 πn

sin (x) = - 3 + 14

使用反三角函数性质

sin (x) = - 3 + 14

sin (x) = - 3 + 14

的通解

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

x = arcsin (- 3 + 14) + 2 πn, x = π - arcsin (- 3 + 14) + 2 πn

x = arcsin (- 3 + 14) + 2 πn, x = π - arcsin (- 3 + 14) + 2 πn

sin (x) = - 3 - 14 :

无解

sin (x) = - 3 - 14

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = arcsin (- 3 + 14) + 2 πn, x = π - arcsin (- 3 + 14) + 2 πn

以小数形式表示解

x = 0.83553 \dots + 2 πn, x = π - 0.83553 \dots + 2 πn

作图

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