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受欢迎的三角函数 >

2cos^2(w)-3cos(w)-5=0

解答

2 cos^{2} (w) - 3 cos (w) - 5 = 0

解答

w = π + 2 πn

+1

度数

w = 18 0^{\circ} + 36 0^{\circ} n

求解步骤

2 cos^{2} (w) - 3 cos (w) - 5 = 0

用替代法求解

2 cos^{2} (w) - 3 cos (w) - 5 = 0

令

: cos (w) = u

2 u^{2} - 3 u - 5 = 0

2 u^{2} - 3 u - 5 = 0 : u = \frac{5}{2}, u = - 1

2 u^{2} - 3 u - 5 = 0

使用求根公式求解

2 u^{2} - 3 u - 5 = 0

二次方程求根公式:

若

a = 2, b = - 3, c = - 5

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 2 ( - 5 )}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 2 ( - 5 )}{2 \cdot 2}

(- 3)^{2} - 4 \cdot 2 (- 5) = 7

(- 3)^{2} - 4 \cdot 2 (- 5)

使用法则

- (- a) = a

= (- 3)^{2} + 4 \cdot 2 \cdot 5

使用指数法则:

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

若

n

是偶数

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

= 3^{2} + 4 \cdot 2 \cdot 5

数字相乘:

4 \cdot 2 \cdot 5 = 40

= 3^{2} + 40

3^{2} = 9

= 9 + 40

数字相加:

9 + 40 = 49

= 49

因式分解数字:

49 = 7^{2}

= 7^{2}

使用根式运算法则:

n a^{n} = a

7^{2} = 7

= 7

u_{1, 2} = \frac{- ( - 3 ) \pm 7}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- ( - 3 ) + 7}{2 \cdot 2}, u_{2} = \frac{- ( - 3 ) - 7}{2 \cdot 2}

u = \frac{- ( - 3 ) + 7}{2 \cdot 2} : \frac{5}{2}

\frac{- ( - 3 ) + 7}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{3 + 7}{2 \cdot 2}

数字相加:

3 + 7 = 10

= \frac{10}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{10}{4}

约分:

2

= \frac{5}{2}

u = \frac{- ( - 3 ) - 7}{2 \cdot 2} : - 1

\frac{- ( - 3 ) - 7}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{3 - 7}{2 \cdot 2}

数字相减:

3 - 7 = - 4

= \frac{- 4}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 4}{4}

使用分式法则:

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

= - \frac{4}{4}

使用法则

\frac{a}{a} = 1

= - 1

二次方程组的解是:

u = \frac{5}{2}, u = - 1

u = cos (w)

代回

cos (w) = \frac{5}{2}, cos (w) = - 1

cos (w) = \frac{5}{2}, cos (w) = - 1

cos (w) = \frac{5}{2} :

无解

cos (w) = \frac{5}{2}

- 1 \leq cos (x) \leq 1

无解

cos (w) = - 1 : w = π + 2 πn

cos (w) = - 1

cos (w) = - 1

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}

w = π + 2 πn

w = π + 2 πn

合并所有解

w = π + 2 πn

作图

流行的例子

4 sin (x) = csc (x)

cot (θ) = - \frac{3}{2}

7sin(x)+sqrt(23)=0

7 sin (x) + 23 = 0

4cos^2(x)+cos(x)-2=0

4 cos^{2} (x) + cos (x) - 2 = 0

tan(x)=-1+sqrt(2)

tan (x) = - 1 + 2