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受欢迎的三角函数 >

cos(x)=1-sin^2(x/2)

解答

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

解答

x = 4 πn, x = 2 π + 4 πn

+1

度数

x = 0^{\circ} + 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n

求解步骤

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

两边减去

1 - sin^{2} (\frac{x}{2})

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

使用三角恒等式改写

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

使用倍角公式:

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

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

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

化简

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

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

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

- (cos (x) - cos^{2} (\frac{x}{2})) : - cos (x) + cos^{2} (\frac{x}{2})

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

打开括号

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

使用加减运算法则

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

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

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

同类项相加:

cos (x) - cos (x) = 0

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

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

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

用替代法求解

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

令

: cos (\frac{x}{2}) = u

- 1 + u^{2} = 0

- 1 + u^{2} = 0 : u = 1, u = - 1

- 1 + u^{2} = 0

将

1

到右边

- 1 + u^{2} = 0

两边加上

1

- 1 + u^{2} + 1 = 0 + 1

化简

u^{2} = 1

u^{2} = 1

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = 1, u = - 1

1 = 1

1

使用法则

1 = 1

= 1

- 1 = - 1

- 1

使用法则

1 = 1

= - 1

u = 1, u = - 1

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

代回

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

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

cos (\frac{x}{2}) = 1 : x = 4 πn

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

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

\frac{x}{2} = 0 + 2 πn

\frac{x}{2} = 0 + 2 πn

解

\frac{x}{2} = 0 + 2 πn : x = 4 πn

\frac{x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{x}{2} = 2 πn

在两边乘以

2

\frac{x}{2} = 2 πn

在两边乘以

2

\frac{2 x}{2} = 2 \cdot 2 πn

化简

x = 4 πn

x = 4 πn

x = 4 πn

cos (\frac{x}{2}) = - 1 : x = 2 π + 4 πn

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

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

\frac{x}{2} = π + 2 πn

\frac{x}{2} = π + 2 πn

解

\frac{x}{2} = π + 2 πn : x = 2 π + 4 πn

\frac{x}{2} = π + 2 πn

在两边乘以

2

\frac{x}{2} = π + 2 πn

在两边乘以

2

\frac{2 x}{2} = 2 π + 2 \cdot 2 πn

化简

x = 2 π + 4 πn

x = 2 π + 4 πn

x = 2 π + 4 πn

合并所有解

x = 4 πn, x = 2 π + 4 πn

作图

流行的例子

3sin(x)-4sin^3(x)=1-2sin^2(x)

3 sin (x) - 4 sin^{3} (x) = 1 - 2 sin^{2} (x)

sin^2(a)=((2tan(a)))/((1+tan^2(a)))

sin^{2} (a) = \frac{( 2 tan ( a ) )}{( 1 + tan ^{2} ( a ) )}

(tan(a))/2 = 2/11

\frac{tan ( a )}{2} = \frac{2}{11}

sin^2(x)-cos(x)= 1/2

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

cos(x)[3sin(x)-2]=0

cos (x) [3 sin (x) - 2] = 0