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受欢迎的三角函数 >

cos(2x)=sin(pi/2-x)

解答

cos (2 x) = sin (\frac{π}{2} - x)

解答

x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

+1

度数

x = 0^{\circ} + 24 0^{\circ} n, x = 12 0^{\circ} + 24 0^{\circ} n

求解步骤

cos (2 x) = sin (\frac{π}{2} - x)

使用三角恒等式改写

cos (2 x) = sin (\frac{π}{2} - x)

使用三角恒等式改写

sin (\frac{π}{2} - x)

使用角差恒等式:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

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

化简

sin (\frac{π}{2}) cos (x) - cos (\frac{π}{2}) sin (x) : cos (x)

sin (\frac{π}{2}) cos (x) - cos (\frac{π}{2}) sin (x)

sin (\frac{π}{2}) cos (x) = cos (x)

sin (\frac{π}{2}) cos (x)

化简

sin (\frac{π}{2}) : 1

sin (\frac{π}{2})

使用以下普通恒等式:

sin (\frac{π}{2}) = 1

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}

= 1

= 1 \cdot cos (x)

乘以:

1 \cdot cos (x) = cos (x)

= cos (x)

cos (\frac{π}{2}) sin (x) = 0

cos (\frac{π}{2}) sin (x)

化简

cos (\frac{π}{2}) : 0

cos (\frac{π}{2})

使用以下普通恒等式:

cos (\frac{π}{2}) = 0

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}

= 0

= 0 \cdot sin (x)

使用法则

0 \cdot a = 0

= 0

= cos (x) - 0

cos (x) - 0 = cos (x)

= cos (x)

= cos (x)

cos (2 x) = cos (x)

cos (2 x) = cos (x)

两边减去

cos (x)

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

使用三角恒等式改写

cos (2 x) - cos (x)

使用和差化积恒等式:

cos (s) - cos (t) = - 2 sin (\frac{s + t}{2}) sin (\frac{s - t}{2})

= - 2 sin (\frac{2 x + x}{2}) sin (\frac{2 x - x}{2})

化简

- 2 sin (\frac{2 x + x}{2}) sin (\frac{2 x - x}{2}) : - 2 sin (\frac{3 x}{2}) sin (\frac{x}{2})

- 2 sin (\frac{2 x + x}{2}) sin (\frac{2 x - x}{2})

同类项相加:

2 x + x = 3 x

= - 2 sin (\frac{3 x}{2}) sin (\frac{2 x - x}{2})

同类项相加:

2 x - x = x

= - 2 sin (\frac{3 x}{2}) sin (\frac{x}{2})

= - 2 sin (\frac{3 x}{2}) sin (\frac{x}{2})

- 2 sin (\frac{3 x}{2}) sin (\frac{x}{2}) = 0

分别求解每个部分

sin (\frac{3 x}{2}) = 0 or sin (\frac{x}{2}) = 0

sin (\frac{3 x}{2}) = 0 : x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

sin (\frac{3 x}{2}) = 0

sin (\frac{3 x}{2}) = 0

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}

\frac{3 x}{2} = 0 + 2 πn, \frac{3 x}{2} = π + 2 πn

\frac{3 x}{2} = 0 + 2 πn, \frac{3 x}{2} = π + 2 πn

解

\frac{3 x}{2} = 0 + 2 πn : x = \frac{4 πn}{3}

\frac{3 x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{3 x}{2} = 2 πn

在两边乘以

2

\frac{3 x}{2} = 2 πn

在两边乘以

2

\frac{2 \cdot 3 x}{2} = 2 \cdot 2 πn

化简

3 x = 4 πn

3 x = 4 πn

两边除以

3

3 x = 4 πn

两边除以

3

\frac{3 x}{3} = \frac{4 πn}{3}

化简

x = \frac{4 πn}{3}

x = \frac{4 πn}{3}

解

\frac{3 x}{2} = π + 2 πn : x = \frac{2 π}{3} + \frac{4 πn}{3}

\frac{3 x}{2} = π + 2 πn

在两边乘以

2

\frac{3 x}{2} = π + 2 πn

在两边乘以

2

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

化简

3 x = 2 π + 4 πn

3 x = 2 π + 4 πn

两边除以

3

3 x = 2 π + 4 πn

两边除以

3

\frac{3 x}{3} = \frac{2 π}{3} + \frac{4 πn}{3}

化简

x = \frac{2 π}{3} + \frac{4 πn}{3}

x = \frac{2 π}{3} + \frac{4 πn}{3}

x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

sin (\frac{x}{2}) = 0 : x = 4 πn, x = 2 π + 4 πn

sin (\frac{x}{2}) = 0

sin (\frac{x}{2}) = 0

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}

\frac{x}{2} = 0 + 2 πn, \frac{x}{2} = π + 2 πn

\frac{x}{2} = 0 + 2 πn, \frac{x}{2} = π + 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

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 = 4 πn, x = 2 π + 4 πn

合并所有解

x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}, x = 4 πn, x = 2 π + 4 πn

合并重叠的区间

x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

作图

流行的例子

sin (x) = sin (5 0^{\circ})

solvefor n,y=-sin(2((3pi)/4+pin))2

so l v e f or n, y = - sin (2 (\frac{3 π}{4} + πn)) 2

tan (θ) = \frac{18}{7}

2cos^2(θ)=1+sin(θ)

2 cos^{2} (θ) = 1 + sin (θ)

3 tan (x) = sin (x)