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受欢迎的三角函数 >

2sin^2(θ)+sqrt(3)sin(θ)=0

解答

2 sin^{2} (θ) + 3 sin (θ) = 0

解答

θ = 2 πn, θ = π + 2 πn, θ = \frac{4 π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

+1

度数

θ = 0^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n, θ = 24 0^{\circ} + 36 0^{\circ} n, θ = 30 0^{\circ} + 36 0^{\circ} n

求解步骤

2 sin^{2} (θ) + 3 sin (θ) = 0

用替代法求解

2 sin^{2} (θ) + 3 sin (θ) = 0

令

: sin (θ) = u

2 u^{2} + 3 u = 0

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

2 u^{2} + 3 u = 0

使用求根公式求解

2 u^{2} + 3 u = 0

二次方程求根公式:

若

a = 2, b = 3, c = 0

u_{1, 2} = \frac{- 3 \pm ( 3 ) ^{2} - 4 \cdot 2 \cdot 0}{2 \cdot 2}

u_{1, 2} = \frac{- 3 \pm ( 3 ) ^{2} - 4 \cdot 2 \cdot 0}{2 \cdot 2}

(3)^{2} - 4 \cdot 2 \cdot 0 = 3

(3)^{2} - 4 \cdot 2 \cdot 0

(3)^{2} = 3

(3)^{2}

使用根式运算法则:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

约分:

2

= 1

= 3

4 \cdot 2 \cdot 0 = 0

4 \cdot 2 \cdot 0

使用法则

0 \cdot a = 0

= 0

= 3 - 0

数字相减:

3 - 0 = 3

= 3

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

将解分隔开

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

u = \frac{- 3 + 3}{2 \cdot 2} : 0

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

同类项相加:

- 3 + 3 = 0

= \frac{0}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{0}{4}

使用法则

\frac{0}{a} = 0, a \neq = 0

= 0

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

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

同类项相加:

- 3 - 3 = - 23

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

数字相乘:

2 \cdot 2 = 4

= \frac{- 2 3}{4}

使用分式法则:

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

= - \frac{2 3}{4}

约分:

2

= - \frac{3}{2}

二次方程组的解是:

u = 0, u = - \frac{3}{2}

u = sin (θ)

代回

sin (θ) = 0, sin (θ) = - \frac{3}{2}

sin (θ) = 0, sin (θ) = - \frac{3}{2}

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

sin (θ) = 0

sin (θ) = 0

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}

θ = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

解

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

sin (θ) = - \frac{3}{2} : θ = \frac{4 π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

sin (θ) = - \frac{3}{2}

sin (θ) = - \frac{3}{2}

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}

θ = \frac{4 π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

θ = \frac{4 π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

合并所有解

θ = 2 πn, θ = π + 2 πn, θ = \frac{4 π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

作图

流行的例子

cos^2(x)=sin^2(x/2)

cos^{2} (x) = sin^{2} (\frac{x}{2})

sin(x+pi/4)=-(sqrt(2))/2

sin (x + \frac{π}{4}) = - \frac{2}{2}

1 = tan (θ)

2 cos (x) - 4 = 0

sec(x)=1-tan(x)

sec (x) = 1 - tan (x)