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受欢迎的三角函数 >

3sin^2(x)-7sin(x)+2=0

解答

3 sin^{2} (x) - 7 sin (x) + 2 = 0

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 = 160.52877 \dots^{\circ} + 36 0^{\circ} n

求解步骤

3 sin^{2} (x) - 7 sin (x) + 2 = 0

用替代法求解

3 sin^{2} (x) - 7 sin (x) + 2 = 0

令

: sin (x) = u

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

3 u^{2} - 7 u + 2 = 0 : u = 2, u = \frac{1}{3}

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

使用求根公式求解

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

二次方程求根公式:

若

a = 3, b = - 7, c = 2

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 3 \cdot 2}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 3 \cdot 2}{2 \cdot 3}

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

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

使用指数法则:

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

若

n

是偶数

(- 7)^{2} = 7^{2}

= 7^{2} - 4 \cdot 3 \cdot 2

数字相乘:

4 \cdot 3 \cdot 2 = 24

= 7^{2} - 24

7^{2} = 49

= 49 - 24

数字相减:

49 - 24 = 25

= 25

因式分解数字:

25 = 5^{2}

= 5^{2}

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 5

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

将解分隔开

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

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

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

使用法则

- (- a) = a

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

数字相加:

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

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

使用法则

- (- a) = a

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

数字相减:

7 - 5 = 2

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

数字相乘:

2 \cdot 3 = 6

= \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

流行的例子

tan (θ) = \frac{4}{3}

tan (x) = \frac{1}{3}

2cos^2(θ)+cos(θ)-1=0

2 cos^{2} (θ) + cos (θ) - 1 = 0

2sin^2(θ)+cos(θ)=1

2 sin^{2} (θ) + cos (θ) = 1

2cos(2x)-2cos(x)=0

2 cos (2 x) - 2 cos (x) = 0