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受欢迎的三角函数 >

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

解答

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

解答

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

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

求解步骤

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

两边减去

1 - 2 sin (2 x)

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

使用三角恒等式改写

- 2 + 2 cos (2 x) + 2 sin (2 x)

使用倍角公式:

sin (2 x) = 2 sin (x) cos (x)

= - 2 + 2 cos (2 x) + 2 \cdot 2 sin (x) cos (x)

化简

= - 2 + 2 cos (2 x) + 4 sin (x) cos (x)

使用倍角公式:

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

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

化简

- 2 + 2 (1 - 2 sin^{2} (x)) + 4 cos (x) sin (x) : 4 cos (x) sin (x) - 4 sin^{2} (x)

- 2 + 2 (1 - 2 sin^{2} (x)) + 4 cos (x) sin (x)

乘开

2 (1 - 2 sin^{2} (x)) : 2 - 4 sin^{2} (x)

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

使用分配律:

a (b - c) = ab - a c

a = 2, b = 1, c = 2 sin^{2} (x)

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

化简

2 \cdot 1 - 2 \cdot 2 sin^{2} (x) : 2 - 4 sin^{2} (x)

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

数字相乘:

2 \cdot 1 = 2

= 2 - 2 \cdot 2 sin^{2} (x)

数字相乘:

2 \cdot 2 = 4

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

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

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

- 2 + 2 = 0

= 4 cos (x) sin (x) - 4 sin^{2} (x)

= 4 cos (x) sin (x) - 4 sin^{2} (x)

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

分解

- 4 sin^{2} (x) + 4 cos (x) sin (x) : 4 sin (x) (- sin (x) + cos (x))

- 4 sin^{2} (x) + 4 cos (x) sin (x)

使用指数法则:

a^{b + c} = a^{b} a^{c}

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

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

因式分解出通项

4 sin (x)

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

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

分别求解每个部分

sin (x) = 0 or - sin (x) + cos (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (x) = 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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

- sin (x) + cos (x) = 0 : x = \frac{π}{4} + πn

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

使用三角恒等式改写

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

在两边除以

cos (x), cos (x) \neq = 0

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

化简

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

使用基本三角恒等式:

\frac{sin ( x )}{cos ( x )} = tan (x)

- tan (x) + 1 = 0

- tan (x) + 1 = 0

将

1

到右边

- tan (x) + 1 = 0

两边减去

1

- tan (x) + 1 - 1 = 0 - 1

化简

- tan (x) = - 1

- tan (x) = - 1

两边除以

- 1

- tan (x) = - 1

两边除以

- 1

\frac{- tan ( x )}{- 1} = \frac{- 1}{- 1}

化简

tan (x) = 1

tan (x) = 1

tan (x) = 1

的通解

tan (x)

周期表（周期为

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

合并所有解

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

作图

流行的例子

cot^2(θ)+3csc(θ)=-3

cot^{2} (θ) + 3 csc (θ) = - 3

3tan^2(α)=3tan(α)

3 tan^{2} (α) = 3 tan (α)

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

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

sin (x) = \frac{10}{50}

6sin(4x)+4=5,0<= x<= 2pi

6 sin (4 x) + 4 = 5, 0 \leq x \leq 2 π