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受欢迎的三角函数 >

5sin^2(x)=2sin(x)

解答

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

解答

x = 2 πn, x = π + 2 πn, x = 0.41151 \dots + 2 πn, x = π - 0.41151 \dots + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 23.57817 \dots^{\circ} + 36 0^{\circ} n, x = 156.42182 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: sin (x) = u

5 u^{2} = 2 u

5 u^{2} = 2 u : u = 0, u = \frac{2}{5}

5 u^{2} = 2 u

将

2 u

para o lado esquerdo

5 u^{2} = 2 u

两边减去

2 u

5 u^{2} - 2 u = 2 u - 2 u

化简

5 u^{2} - 2 u = 0

5 u^{2} - 2 u = 0

因式分解

5 u^{2} - 2 u : u (5 u - 2)

5 u^{2} - 2 u

使用指数法则:

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

u^{2} = uu

= 5 uu - 2 u

因式分解出通项

u

= u (5 u - 2)

u (5 u - 2) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

u = 0 or 5 u - 2 = 0

解

5 u - 2 = 0 : u = \frac{2}{5}

5 u - 2 = 0

将

2

到右边

5 u - 2 = 0

两边加上

2

5 u - 2 + 2 = 0 + 2

化简

5 u = 2

5 u = 2

两边除以

5

5 u = 2

两边除以

5

\frac{5 u}{5} = \frac{2}{5}

化简

u = \frac{2}{5}

u = \frac{2}{5}

二次方程组的解是:

u = 0, u = \frac{2}{5}

u = sin (x)

代回

sin (x) = 0, sin (x) = \frac{2}{5}

sin (x) = 0, sin (x) = \frac{2}{5}

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) = \frac{2}{5} : x = arcsin (\frac{2}{5}) + 2 πn, x = π - arcsin (\frac{2}{5}) + 2 πn

sin (x) = \frac{2}{5}

使用反三角函数性质

sin (x) = \frac{2}{5}

sin (x) = \frac{2}{5}

的通解

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

x = arcsin (\frac{2}{5}) + 2 πn, x = π - arcsin (\frac{2}{5}) + 2 πn

x = arcsin (\frac{2}{5}) + 2 πn, x = π - arcsin (\frac{2}{5}) + 2 πn

合并所有解

x = 2 πn, x = π + 2 πn, x = arcsin (\frac{2}{5}) + 2 πn, x = π - arcsin (\frac{2}{5}) + 2 πn

以小数形式表示解

x = 2 πn, x = π + 2 πn, x = 0.41151 \dots + 2 πn, x = π - 0.41151 \dots + 2 πn

作图

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