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受欢迎的三角函数 >

3sin^2(x)+2sin(x)-1=0

解答

3 sin^{2} (x) + 2 sin (x) - 1 = 0

解答

x = 0.33983 \dots + 2 πn, x = π - 0.33983 \dots + 2 πn, x = \frac{3 π}{2} + 2 πn

+1

度数

x = 19.47122 \dots^{\circ} + 36 0^{\circ} n, x = 160.52877 \dots^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

求解步骤

3 sin^{2} (x) + 2 sin (x) - 1 = 0

用替代法求解

3 sin^{2} (x) + 2 sin (x) - 1 = 0

令

: sin (x) = u

3 u^{2} + 2 u - 1 = 0

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

3 u^{2} + 2 u - 1 = 0

使用求根公式求解

3 u^{2} + 2 u - 1 = 0

二次方程求根公式:

若

a = 3, b = 2, c = - 1

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

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

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

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

使用法则

- (- a) = a

= 2^{2} + 4 \cdot 3 \cdot 1

数字相乘:

4 \cdot 3 \cdot 1 = 12

= 2^{2} + 12

2^{2} = 4

= 4 + 12

数字相加:

4 + 12 = 16

= 16

因式分解数字:

16 = 4^{2}

= 4^{2}

使用根式运算法则:

n a^{n} = a

4^{2} = 4

= 4

u_{1, 2} = \frac{- 2 \pm 4}{2 \cdot 3}

将解分隔开

u_{1} = \frac{- 2 + 4}{2 \cdot 3}, u_{2} = \frac{- 2 - 4}{2 \cdot 3}

u = \frac{- 2 + 4}{2 \cdot 3} : \frac{1}{3}

\frac{- 2 + 4}{2 \cdot 3}

数字相加/相减:

- 2 + 4 = 2

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

数字相乘:

2 \cdot 3 = 6

= \frac{2}{6}

约分:

2

= \frac{1}{3}

u = \frac{- 2 - 4}{2 \cdot 3} : - 1

\frac{- 2 - 4}{2 \cdot 3}

数字相减:

- 2 - 4 = - 6

= \frac{- 6}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{- 6}{6}

使用分式法则:

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

= - \frac{6}{6}

使用法则

\frac{a}{a} = 1

= - 1

二次方程组的解是:

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

u = sin (x)

代回

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

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

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

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

sin (x) = - 1

sin (x) = - 1

的通解

sin (x)

周期表（周期为

2 πn

"）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

合并所有解

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

以小数形式表示解

x = 0.33983 \dots + 2 πn, x = π - 0.33983 \dots + 2 πn, x = \frac{3 π}{2} + 2 πn

作图

流行的例子

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2 cos (θ) = 3

-7sqrt(3)tan(θ)+2=9

- 73 tan (θ) + 2 = 9

tan^2(x)+2sec(x)=-1

tan^{2} (x) + 2 sec (x) = - 1

2cos^2(θ)+3cos(θ)+1=0

2 cos^{2} (θ) + 3 cos (θ) + 1 = 0

tan^2(x)sin(x)=-sin(x)

tan^{2} (x) sin (x) = - sin (x)