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受欢迎的三角函数 >

2sin^2(x)-sqrt(2)=0

解答

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

解答

x = 0.99893 \dots + 2 πn, x = π - 0.99893 \dots + 2 πn, x = - 0.99893 \dots + 2 πn, x = π + 0.99893 \dots + 2 πn

+1

度数

x = 57.23490 \dots^{\circ} + 36 0^{\circ} n, x = 122.76509 \dots^{\circ} + 36 0^{\circ} n, x = - 57.23490 \dots^{\circ} + 36 0^{\circ} n, x = 237.23490 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: sin (x) = u

2 u^{2} - 2 = 0

2 u^{2} - 2 = 0 : u = \frac{2}{2}, u = - \frac{2}{2}

2 u^{2} - 2 = 0

将

2

到右边

2 u^{2} - 2 = 0

两边加上

2

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

化简

2 u^{2} = 2

2 u^{2} = 2

两边除以

2

2 u^{2} = 2

两边除以

2

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

化简

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

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

对于

x^{2} = f (a)

解为

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

u = \frac{2}{2}, u = - \frac{2}{2}

u = sin (x)

代回

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

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

sin (x) = \frac{2}{2} : x = arcsin \frac{2}{2} + 2 πn, x = π - arcsin \frac{2}{2} + 2 πn

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

使用反三角函数性质

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

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

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin \frac{2}{2} + 2 πn, x = π - arcsin \frac{2}{2} + 2 πn

x = arcsin \frac{2}{2} + 2 πn, x = π - arcsin \frac{2}{2} + 2 πn

sin (x) = - \frac{2}{2} : x = arcsin - \frac{2}{2} + 2 πn, x = π + arcsin \frac{2}{2} + 2 πn

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

使用反三角函数性质

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

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

的通解

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin - \frac{2}{2} + 2 πn, x = π + arcsin \frac{2}{2} + 2 πn

x = arcsin - \frac{2}{2} + 2 πn, x = π + arcsin \frac{2}{2} + 2 πn

合并所有解

x = arcsin \frac{2}{2} + 2 πn, x = π - arcsin \frac{2}{2} + 2 πn, x = arcsin - \frac{2}{2} + 2 πn, x = π + arcsin \frac{2}{2} + 2 πn

以小数形式表示解

x = 0.99893 \dots + 2 πn, x = π - 0.99893 \dots + 2 πn, x = - 0.99893 \dots + 2 πn, x = π + 0.99893 \dots + 2 πn

作图

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