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受欢迎的三角函数 >

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

解答

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

解答

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

+1

度数

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

cos (2 x)

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

使用三角恒等式改写

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

使用倍角公式:

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

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

化简

- 2 - (1 - 2 sin^{2} (x)) + sin^{2} (x) : 3 sin^{2} (x) - 3

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

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

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

打开括号

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

使用加减运算法则

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

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

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

化简

- 2 - 1 + 2 sin^{2} (x) + sin^{2} (x) : 3 sin^{2} (x) - 3

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

同类项相加:

2 sin^{2} (x) + sin^{2} (x) = 3 sin^{2} (x)

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

数字相减:

- 2 - 1 = - 3

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

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

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

- 3 + 3 sin^{2} (x) = 0

用替代法求解

- 3 + 3 sin^{2} (x) = 0

令

: sin (x) = u

- 3 + 3 u^{2} = 0

- 3 + 3 u^{2} = 0 : u = 1, u = - 1

- 3 + 3 u^{2} = 0

将

3

到右边

- 3 + 3 u^{2} = 0

两边加上

3

- 3 + 3 u^{2} + 3 = 0 + 3

化简

3 u^{2} = 3

3 u^{2} = 3

两边除以

3

3 u^{2} = 3

两边除以

3

\frac{3 u ^{2}}{3} = \frac{3}{3}

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 = sin (x)

代回

sin (x) = 1, sin (x) = - 1

sin (x) = 1, sin (x) = - 1

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = 1

sin (x) = 1

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 = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

sin (x) = - 1 : x = \frac{3 π}{2} + 2 πn

sin (x) = - 1

sin (x) = - 1

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 = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

合并所有解

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

作图

流行的例子

3 + 6 sin (3 x) = 0

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

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

solvefor θ,x=5tan(θ)

so l v e f or θ, x = 5 tan (θ)

3sin(x)+5cos(x)+5=0

3 sin (x) + 5 cos (x) + 5 = 0

4 cos^{2} (x) + 2 = 3