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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{π}{2} + 2 πn

+1

度数

θ = 9 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 = \frac{7}{6}, u = 1

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 ( - 13 ) ^{2} - 4 \cdot 6 \cdot 7}{2 \cdot 6}

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

(- 13)^{2} - 4 \cdot 6 \cdot 7 = 1

(- 13)^{2} - 4 \cdot 6 \cdot 7

使用指数法则:

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

若

n

是偶数

(- 13)^{2} = 1 3^{2}

= 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} : \frac{7}{6}

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

使用法则

- (- a) = a

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

数字相加:

13 + 1 = 14

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

数字相乘:

2 \cdot 6 = 12

= \frac{14}{12}

约分:

2

= \frac{7}{6}

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

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

使用法则

- (- a) = a

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

数字相减:

13 - 1 = 12

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

数字相乘:

2 \cdot 6 = 12

= \frac{12}{12}

使用法则

\frac{a}{a} = 1

= 1

二次方程组的解是:

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

u = sin (θ)

代回

sin (θ) = \frac{7}{6}, sin (θ) = 1

sin (θ) = \frac{7}{6}, sin (θ) = 1

sin (θ) = \frac{7}{6} :

无解

sin (θ) = \frac{7}{6}

- 1 \leq sin (x) \leq 1

无解

sin (θ) = 1 : θ = \frac{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{π}{2} + 2 πn

θ = \frac{π}{2} + 2 πn

合并所有解

θ = \frac{π}{2} + 2 πn

作图

流行的例子

sin (x) = 0.1

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

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

3sin(x)-4=sin(x)-2

3 sin (x) - 4 = sin (x) - 2

sin (θ) = - \frac{5}{6}

sec (θ) = 0