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受欢迎的三角函数 >

sin^2(x)+5/14 = 6/7

解答

sin^{2} (x) + \frac{5}{14} = \frac{6}{7}

解答

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

+1

度数

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

求解步骤

sin^{2} (x) + \frac{5}{14} = \frac{6}{7}

用替代法求解

sin^{2} (x) + \frac{5}{14} = \frac{6}{7}

令

: sin (x) = u

u^{2} + \frac{5}{14} = \frac{6}{7}

u^{2} + \frac{5}{14} = \frac{6}{7} : u = \frac{1}{2}, u = - \frac{1}{2}

u^{2} + \frac{5}{14} = \frac{6}{7}

将

\frac{5}{14}

到右边

u^{2} + \frac{5}{14} = \frac{6}{7}

两边减去

\frac{5}{14}

u^{2} + \frac{5}{14} - \frac{5}{14} = \frac{6}{7} - \frac{5}{14}

化简

u^{2} = \frac{6}{7} - \frac{5}{14}

u^{2} = \frac{6}{7} - \frac{5}{14}

化简

\frac{6}{7} - \frac{5}{14} : \frac{1}{2}

\frac{6}{7} - \frac{5}{14}

7, 14

的最小公倍数:

14

7, 14

最小公倍数 (LCM)

7

质因数分解:

7

7

7

是质数，因此无法因数分解

= 7

14

质因数分解:

2 \cdot 7

14

14

除以

214 = 7 \cdot 2

= 2 \cdot 7

2, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 7

将每个因子乘以它在

7

或

14

中出现的最多次数

= 7 \cdot 2

数字相乘:

7 \cdot 2 = 14

= 14

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

14

对于

\frac{6}{7} :

将分母和分子乘以

2

\frac{6}{7} = \frac{6 \cdot 2}{7 \cdot 2} = \frac{12}{14}

= \frac{12}{14} - \frac{5}{14}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{12 - 5}{14}

数字相减:

12 - 5 = 7

= \frac{7}{14}

约分:

7

= \frac{1}{2}

对于

x^{2} = f (a)

解为

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

u = \frac{1}{2}, u = - \frac{1}{2}

u = sin (x)

代回

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = \frac{1}{2} : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2}

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}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) = - \frac{1}{2} : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sin (x) = - \frac{1}{2}

sin (x) = - \frac{1}{2}

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{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

合并所有解

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

作图

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