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受欢迎的三角函数 >

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

解答

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

解答

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

+1

度数

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

求解步骤

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

1 = cos^{2} (x) + sin^{2} (x)

1 - sin^{2} (x) = cos^{2} (x)

= - 3 cos (x) - cos^{2} (x)

- cos^{2} (x) - 3 cos (x) = 0

用替代法求解

- cos^{2} (x) - 3 cos (x) = 0

令

: cos (x) = u

- u^{2} - 3 u = 0

- u^{2} - 3 u = 0 : u = - 3, u = 0

- u^{2} - 3 u = 0

使用求根公式求解

- u^{2} - 3 u = 0

二次方程求根公式:

若

a = - 1, b = - 3, c = 0

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

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

(- 3)^{2} - 4 (- 1) \cdot 0 = 3

(- 3)^{2} - 4 (- 1) \cdot 0

使用法则

- (- a) = a

= (- 3)^{2} + 4 \cdot 1 \cdot 0

使用指数法则:

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

若

n

是偶数

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

= 3^{2} + 4 \cdot 1 \cdot 0

使用法则

0 \cdot a = 0

= 3^{2} + 0

3^{2} + 0 = 3^{2}

= 3^{2}

使用根式运算法则:

n a^{n} = a,

假定

a \geq 0

= 3

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

将解分隔开

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

u = \frac{- ( - 3 ) + 3}{2 ( - 1 )} : - 3

\frac{- ( - 3 ) + 3}{2 ( - 1 )}

去除括号:

(- a) = - a, - (- a) = a

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

数字相加:

3 + 3 = 6

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

数字相乘:

2 \cdot 1 = 2

= \frac{6}{- 2}

使用分式法则:

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

= - \frac{6}{2}

数字相除:

\frac{6}{2} = 3

= - 3

u = \frac{- ( - 3 ) - 3}{2 ( - 1 )} : 0

\frac{- ( - 3 ) - 3}{2 ( - 1 )}

去除括号:

(- a) = - a, - (- a) = a

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

数字相减:

3 - 3 = 0

= \frac{0}{- 2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{0}{- 2}

使用分式法则:

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

= - \frac{0}{2}

使用法则

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

= - 0

= 0

二次方程组的解是:

u = - 3, u = 0

u = cos (x)

代回

cos (x) = - 3, cos (x) = 0

cos (x) = - 3, cos (x) = 0

cos (x) = - 3 :

无解

cos (x) = - 3

- 1 \leq cos (x) \leq 1

无解

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

cos (x) = 0

的通解

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

合并所有解

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

作图

流行的例子

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