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受欢迎的三角函数 >

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

解答

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

解答

x = 2 πn, x = π + 2 πn

+1

度数

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

求解步骤

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

两边减去

1 - sin^{2} (x)

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

使用三角恒等式改写

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

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

化简

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

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

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

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

打开括号

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

使用加减运算法则

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

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

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

同类项相加:

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

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

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

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

用替代法求解

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

令

: cos (x) = 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 (x)

代回

cos (x) = 1, cos (x) = - 1

cos (x) = 1, cos (x) = - 1

cos (x) = 1 : x = 2 πn

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 = 0 + 2 πn

x = 0 + 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

cos (x) = - 1 : x = π + 2 πn

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

合并所有解

x = 2 πn, x = π + 2 πn

作图

流行的例子

tan (x) = 1.35

arctan(x+1x-1)+arctan(x-12)=arctan(2)

arctan (x + 1 x - 1) + arctan (x - 12) = arctan (2)

sin (α) = \frac{1}{4}

3cos^2(3x)-9/4 =0

3 cos^{2} (3 x) - \frac{9}{4} = 0

-5cos(x)+8.6602500000000…sin(x)=-5

- 5 cos (x) + 8.6602500000000 \dots sin (x) = - 5