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受欢迎的三角函数 >

7sin^2(θ)-16sin(θ)+9=0

解答

7 sin^{2} (θ) - 16 sin (θ) + 9 = 0

解答

θ = \frac{π}{2} + 2 πn

+1

度数

θ = 9 0^{\circ} + 36 0^{\circ} n

求解步骤

7 sin^{2} (θ) - 16 sin (θ) + 9 = 0

用替代法求解

7 sin^{2} (θ) - 16 sin (θ) + 9 = 0

令

: sin (θ) = u

7 u^{2} - 16 u + 9 = 0

7 u^{2} - 16 u + 9 = 0 : u = \frac{9}{7}, u = 1

7 u^{2} - 16 u + 9 = 0

使用求根公式求解

7 u^{2} - 16 u + 9 = 0

二次方程求根公式:

若

a = 7, b = - 16, c = 9

u_{1, 2} = \frac{- ( - 16 ) \pm ( - 16 ) ^{2} - 4 \cdot 7 \cdot 9}{2 \cdot 7}

u_{1, 2} = \frac{- ( - 16 ) \pm ( - 16 ) ^{2} - 4 \cdot 7 \cdot 9}{2 \cdot 7}

(- 16)^{2} - 4 \cdot 7 \cdot 9 = 2

(- 16)^{2} - 4 \cdot 7 \cdot 9

使用指数法则:

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

若

n

是偶数

(- 16)^{2} = 1 6^{2}

= 1 6^{2} - 4 \cdot 7 \cdot 9

数字相乘:

4 \cdot 7 \cdot 9 = 252

= 1 6^{2} - 252

1 6^{2} = 256

= 256 - 252

数字相减:

256 - 252 = 4

= 4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

u_{1, 2} = \frac{- ( - 16 ) \pm 2}{2 \cdot 7}

将解分隔开

u_{1} = \frac{- ( - 16 ) + 2}{2 \cdot 7}, u_{2} = \frac{- ( - 16 ) - 2}{2 \cdot 7}

u = \frac{- ( - 16 ) + 2}{2 \cdot 7} : \frac{9}{7}

\frac{- ( - 16 ) + 2}{2 \cdot 7}

使用法则

- (- a) = a

= \frac{16 + 2}{2 \cdot 7}

数字相加:

16 + 2 = 18

= \frac{18}{2 \cdot 7}

数字相乘:

2 \cdot 7 = 14

= \frac{18}{14}

约分:

2

= \frac{9}{7}

u = \frac{- ( - 16 ) - 2}{2 \cdot 7} : 1

\frac{- ( - 16 ) - 2}{2 \cdot 7}

使用法则

- (- a) = a

= \frac{16 - 2}{2 \cdot 7}

数字相减:

16 - 2 = 14

= \frac{14}{2 \cdot 7}

数字相乘:

2 \cdot 7 = 14

= \frac{14}{14}

使用法则

\frac{a}{a} = 1

= 1

二次方程组的解是:

u = \frac{9}{7}, u = 1

u = sin (θ)

代回

sin (θ) = \frac{9}{7}, sin (θ) = 1

sin (θ) = \frac{9}{7}, sin (θ) = 1

sin (θ) = \frac{9}{7} :

无解

sin (θ) = \frac{9}{7}

- 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

cos^2(x)-cos(x)-6=0

cos^{2} (x) - cos (x) - 6 = 0

8cos(x)tan(x)=3tan(x)

8 cos (x) tan (x) = 3 tan (x)

cos(x/2)=-(sqrt(2))/2

cos (\frac{x}{2}) = - \frac{2}{2}

0=sqrt(1-cos(2x))

0 = 1 - cos (2 x)

sec^2(x)+2=3sec(x)

sec^{2} (x) + 2 = 3 sec (x)