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受欢迎的三角函数 >

sin^2(x)-6sin(x)+3=0

解答

sin^{2} (x) - 6 sin (x) + 3 = 0

解答

x = 0.58297 \dots + 2 πn, x = π - 0.58297 \dots + 2 πn

+1

度数

x = 33.40202 \dots^{\circ} + 36 0^{\circ} n, x = 146.59797 \dots^{\circ} + 36 0^{\circ} n

求解步骤

sin^{2} (x) - 6 sin (x) + 3 = 0

用替代法求解

sin^{2} (x) - 6 sin (x) + 3 = 0

令

: sin (x) = u

u^{2} - 6 u + 3 = 0

u^{2} - 6 u + 3 = 0 : u = 3 + 6, u = 3 - 6

u^{2} - 6 u + 3 = 0

使用求根公式求解

u^{2} - 6 u + 3 = 0

二次方程求根公式:

若

a = 1, b = - 6, c = 3

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 1 \cdot 3}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 1 \cdot 3}{2 \cdot 1}

(- 6)^{2} - 4 \cdot 1 \cdot 3 = 26

(- 6)^{2} - 4 \cdot 1 \cdot 3

使用指数法则:

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

若

n

是偶数

(- 6)^{2} = 6^{2}

= 6^{2} - 4 \cdot 1 \cdot 3

数字相乘:

4 \cdot 1 \cdot 3 = 12

= 6^{2} - 12

6^{2} = 36

= 36 - 12

数字相减:

36 - 12 = 24

= 24

24

质因数分解:

2^{3} \cdot 3

24

24

除以

224 = 12 \cdot 2

= 2 \cdot 12

12

除以

212 = 6 \cdot 2

= 2 \cdot 2 \cdot 6

6

除以

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2 \cdot 3

= 2^{3} \cdot 3

= 2^{3} \cdot 3

使用指数法则:

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

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

使用根式运算法则:

n ab = n a n b

= 2^{2} 2 \cdot 3

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2 2 \cdot 3

整理后得

= 26

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

将解分隔开

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

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

\frac{- ( - 6 ) + 2 6}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{6 + 2 6}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

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

分解

6 + 26 : 2 (3 + 6)

6 + 26

改写为

= 2 \cdot 3 + 26

因式分解出通项

2

= 2 (3 + 6)

= \frac{2 ( 3 + 6 )}{2}

数字相除:

\frac{2}{2} = 1

= 3 + 6

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

\frac{- ( - 6 ) - 2 6}{2 \cdot 1}

使用法则

- (- a) = a

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

数字相乘:

2 \cdot 1 = 2

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

分解

6 - 26 : 2 (3 - 6)

6 - 26

改写为

= 2 \cdot 3 - 26

因式分解出通项

2

= 2 (3 - 6)

= \frac{2 ( 3 - 6 )}{2}

数字相除:

\frac{2}{2} = 1

= 3 - 6

二次方程组的解是:

u = 3 + 6, u = 3 - 6

u = sin (x)

代回

sin (x) = 3 + 6, sin (x) = 3 - 6

sin (x) = 3 + 6, sin (x) = 3 - 6

sin (x) = 3 + 6 :

无解

sin (x) = 3 + 6

- 1 \leq sin (x) \leq 1

无解

sin (x) = 3 - 6 : x = arcsin (3 - 6) + 2 πn, x = π - arcsin (3 - 6) + 2 πn

sin (x) = 3 - 6

使用反三角函数性质

sin (x) = 3 - 6

sin (x) = 3 - 6

的通解

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

x = arcsin (3 - 6) + 2 πn, x = π - arcsin (3 - 6) + 2 πn

x = arcsin (3 - 6) + 2 πn, x = π - arcsin (3 - 6) + 2 πn

合并所有解

x = arcsin (3 - 6) + 2 πn, x = π - arcsin (3 - 6) + 2 πn

以小数形式表示解

x = 0.58297 \dots + 2 πn, x = π - 0.58297 \dots + 2 πn

作图

流行的例子

12csc^2(θ)-csc(θ)-1=0

12 csc^{2} (θ) - csc (θ) - 1 = 0

2cos(2θ)-4cos(θ)+2=-1

2 cos (2 θ) - 4 cos (θ) + 2 = - 1

cos(a)= 1/(sin(a))

cos (a) = \frac{1}{sin ( a )}

sin (a) = cos (a)

tan (a) = 2.0503