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受欢迎的三角函数 >

6cos^2(x/2)+cos(x)+1=0

解答

6 cos^{2} (\frac{x}{2}) + cos (x) + 1 = 0

解答

x = π + 2 πn

+1

度数

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

求解步骤

6 cos^{2} (\frac{x}{2}) + cos (x) + 1 = 0

使用三角恒等式改写

1 + cos (x) + 6 cos^{2} (\frac{x}{2})

利用以下特性:

cos^{2} (θ) = \frac{1 + cos ( 2 θ )}{2}

使用倍角公式

cos (2 θ) = 2 cos^{2} (θ) - 1

交换两边

2 cos^{2} (θ) - 1 = cos (2 θ)

两边加上

1

2 sin^{2} (θ) = 1 + cos (2 θ)

两边除以

2

cos^{2} (θ) = \frac{1 + cos ( 2 θ )}{2}

= 1 + cos (x) + 6 \cdot \frac{1 + cos ( 2 \cdot \frac{x}{2} )}{2}

化简

1 + cos (x) + 6 \cdot \frac{1 + cos ( 2 \cdot \frac{x}{2} )}{2} : 4 cos (x) + 4

1 + cos (x) + 6 \cdot \frac{1 + cos ( 2 \cdot \frac{x}{2} )}{2}

6 \cdot \frac{1 + cos ( 2 \cdot \frac{x}{2} )}{2} = 3 (cos (x) + 1)

6 \cdot \frac{1 + cos ( 2 \cdot \frac{x}{2} )}{2}

\frac{1 + cos ( 2 \cdot \frac{x}{2} )}{2} = \frac{1 + cos ( x )}{2}

\frac{1 + cos ( 2 \cdot \frac{x}{2} )}{2}

乘

2 \cdot \frac{x}{2} : x

2 \cdot \frac{x}{2}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{x \cdot 2}{2}

约分:

2

= x

= \frac{1 + cos ( x )}{2}

= 6 \cdot \frac{cos ( x ) + 1}{2}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( 1 + cos ( x ) ) \cdot 6}{2}

数字相除:

\frac{6}{2} = 3

= 3 (cos (x) + 1)

= 1 + cos (x) + 3 (cos (x) + 1)

乘开

3 (cos (x) + 1) : 3 cos (x) + 3

3 (cos (x) + 1)

使用分配律:

a (b + c) = ab + a c

a = 3, b = cos (x), c = 1

= 3 cos (x) + 3 \cdot 1

数字相乘:

3 \cdot 1 = 3

= 3 cos (x) + 3

= 1 + cos (x) + 3 cos (x) + 3

化简

1 + cos (x) + 3 cos (x) + 3 : 4 cos (x) + 4

1 + cos (x) + 3 cos (x) + 3

同类项相加:

cos (x) + 3 cos (x) = 4 cos (x)

= 1 + 4 cos (x) + 3

对同类项分组

= 4 cos (x) + 1 + 3

数字相加:

1 + 3 = 4

= 4 cos (x) + 4

= 4 cos (x) + 4

= 4 cos (x) + 4

4 + 4 cos (x) = 0

将

4

到右边

4 + 4 cos (x) = 0

两边减去

4

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

化简

4 cos (x) = - 4

4 cos (x) = - 4

两边除以

4

4 cos (x) = - 4

两边除以

4

\frac{4 cos ( x )}{4} = \frac{- 4}{4}

化简

cos (x) = - 1

cos (x) = - 1

cos (x) = - 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}

x = π + 2 πn

x = π + 2 πn

作图

流行的例子

- 2 sin (θ) = 1

cos (θ) = - \frac{7}{3}

0=-6sin(x)+6cos(2x)

0 = - 6 sin (x) + 6 cos (2 x)

tan (2 x + 1) = \frac{1}{2}

cos (B) = \frac{8}{12}