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受欢迎的三角函数 >

6sin^2(θ)+13sin(θ)+7=0

解答

6 sin^{2} (θ) + 13 sin (θ) + 7 = 0

解答

θ = \frac{3 π}{2} + 2 πn

+1

度数

θ = 27 0^{\circ} + 36 0^{\circ} n

求解步骤

6 sin^{2} (θ) + 13 sin (θ) + 7 = 0

用替代法求解

6 sin^{2} (θ) + 13 sin (θ) + 7 = 0

令

: sin (θ) = u

6 u^{2} + 13 u + 7 = 0

6 u^{2} + 13 u + 7 = 0 : u = - 1, u = - \frac{7}{6}

6 u^{2} + 13 u + 7 = 0

使用求根公式求解

6 u^{2} + 13 u + 7 = 0

二次方程求根公式:

若

a = 6, b = 13, c = 7

u_{1, 2} = \frac{- 13 \pm 1 3 ^{2} - 4 \cdot 6 \cdot 7}{2 \cdot 6}

u_{1, 2} = \frac{- 13 \pm 1 3 ^{2} - 4 \cdot 6 \cdot 7}{2 \cdot 6}

1 3^{2} - 4 \cdot 6 \cdot 7 = 1

1 3^{2} - 4 \cdot 6 \cdot 7

数字相乘:

4 \cdot 6 \cdot 7 = 168

= 1 3^{2} - 168

1 3^{2} = 169

= 169 - 168

数字相减:

169 - 168 = 1

= 1

使用法则

1 = 1

= 1

u_{1, 2} = \frac{- 13 \pm 1}{2 \cdot 6}

将解分隔开

u_{1} = \frac{- 13 + 1}{2 \cdot 6}, u_{2} = \frac{- 13 - 1}{2 \cdot 6}

u = \frac{- 13 + 1}{2 \cdot 6} : - 1

\frac{- 13 + 1}{2 \cdot 6}

数字相加/相减:

- 13 + 1 = - 12

= \frac{- 12}{2 \cdot 6}

数字相乘:

2 \cdot 6 = 12

= \frac{- 12}{12}

使用分式法则:

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

= - \frac{12}{12}

使用法则

\frac{a}{a} = 1

= - 1

u = \frac{- 13 - 1}{2 \cdot 6} : - \frac{7}{6}

\frac{- 13 - 1}{2 \cdot 6}

数字相减:

- 13 - 1 = - 14

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

数字相乘:

2 \cdot 6 = 12

= \frac{- 14}{12}

使用分式法则:

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

= - \frac{14}{12}

约分:

2

= - \frac{7}{6}

二次方程组的解是:

u = - 1, u = - \frac{7}{6}

u = sin (θ)

代回

sin (θ) = - 1, sin (θ) = - \frac{7}{6}

sin (θ) = - 1, sin (θ) = - \frac{7}{6}

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

sin (θ) = - 1

sin (θ) = - 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}

θ = \frac{3 π}{2} + 2 πn

θ = \frac{3 π}{2} + 2 πn

sin (θ) = - \frac{7}{6} :

无解

sin (θ) = - \frac{7}{6}

- 1 \leq sin (x) \leq 1

无解

合并所有解

θ = \frac{3 π}{2} + 2 πn

作图

流行的例子

12sin^2(x)-17sin(x)+6=0

12 sin^{2} (x) - 17 sin (x) + 6 = 0

sin(θ)=(sqrt(3))/3

sin (θ) = \frac{3}{3}

tan(x+pi)+cos(x+pi/2)=0

tan (x + π) + cos (x + \frac{π}{2}) = 0

sin(θ)cos(θ)=(sqrt(3))/4

sin (θ) cos (θ) = \frac{3}{4}

tan(x+pi/6)=sqrt(3)

tan (x + \frac{π}{6}) = 3