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受欢迎的三角函数 >

7sin(2x)+12sin(x)=0

解答

7 sin (2 x) + 12 sin (x) = 0

解答

x = 2 πn, x = π + 2 πn, x = 2.60049 \dots + 2 πn, x = - 2.60049 \dots + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 148.99728 \dots^{\circ} + 36 0^{\circ} n, x = - 148.99728 \dots^{\circ} + 36 0^{\circ} n

求解步骤

7 sin (2 x) + 12 sin (x) = 0

使用三角恒等式改写

12 sin (x) + 7 sin (2 x)

使用倍角公式:

sin (2 x) = 2 sin (x) cos (x)

= 12 sin (x) + 7 \cdot 2 sin (x) cos (x)

化简

= 12 sin (x) + 14 sin (x) cos (x)

12 sin (x) + 14 cos (x) sin (x) = 0

分解

12 sin (x) + 14 cos (x) sin (x) : 2 sin (x) (7 cos (x) + 6)

12 sin (x) + 14 cos (x) sin (x)

将

14

改写为

7 \cdot 2

将

12

改写为

6 \cdot 2

= 6 \cdot 2 sin (x) + 7 \cdot 2 sin (x) cos (x)

因式分解出通项

2 sin (x)

= 2 sin (x) (6 + 7 cos (x))

2 sin (x) (7 cos (x) + 6) = 0

分别求解每个部分

sin (x) = 0 or 7 cos (x) + 6 = 0

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

7 cos (x) + 6 = 0 : x = arccos (- \frac{6}{7}) + 2 πn, x = - arccos (- \frac{6}{7}) + 2 πn

7 cos (x) + 6 = 0

将

6

到右边

7 cos (x) + 6 = 0

两边减去

6

7 cos (x) + 6 - 6 = 0 - 6

化简

7 cos (x) = - 6

7 cos (x) = - 6

两边除以

7

7 cos (x) = - 6

两边除以

7

\frac{7 cos ( x )}{7} = \frac{- 6}{7}

化简

cos (x) = - \frac{6}{7}

cos (x) = - \frac{6}{7}

使用反三角函数性质

cos (x) = - \frac{6}{7}

cos (x) = - \frac{6}{7}

的通解

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- \frac{6}{7}) + 2 πn, x = - arccos (- \frac{6}{7}) + 2 πn

x = arccos (- \frac{6}{7}) + 2 πn, x = - arccos (- \frac{6}{7}) + 2 πn

合并所有解

x = 2 πn, x = π + 2 πn, x = arccos (- \frac{6}{7}) + 2 πn, x = - arccos (- \frac{6}{7}) + 2 πn

以小数形式表示解

x = 2 πn, x = π + 2 πn, x = 2.60049 \dots + 2 πn, x = - 2.60049 \dots + 2 πn

作图

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