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受欢迎的三角函数 >

-6sin(x)-4sin(2x)=0

解答

- 6 sin (x) - 4 sin (2 x) = 0

解答

x = 2 πn, x = π + 2 πn, x = 2.41885 \dots + 2 πn, x = - 2.41885 \dots + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 138.59037 \dots^{\circ} + 36 0^{\circ} n, x = - 138.59037 \dots^{\circ} + 36 0^{\circ} n

求解步骤

- 6 sin (x) - 4 sin (2 x) = 0

使用三角恒等式改写

- 4 sin (2 x) - 6 sin (x)

使用倍角公式:

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

= - 4 \cdot 2 sin (x) cos (x) - 6 sin (x)

化简

= - 8 sin (x) cos (x) - 6 sin (x)

- 6 sin (x) - 8 cos (x) sin (x) = 0

分解

- 6 sin (x) - 8 cos (x) sin (x) : - 2 sin (x) (4 cos (x) + 3)

- 6 sin (x) - 8 cos (x) sin (x)

将

- 8

改写为

4 \cdot 2

将

- 6

改写为

3 \cdot 2

= 3 \cdot 2 sin (x) + 4 \cdot 2 sin (x) cos (x)

因式分解出通项

- 2 sin (x)

= - 2 sin (x) (3 + 4 cos (x))

- 2 sin (x) (4 cos (x) + 3) = 0

分别求解每个部分

sin (x) = 0 or 4 cos (x) + 3 = 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

4 cos (x) + 3 = 0 : x = arccos (- \frac{3}{4}) + 2 πn, x = - arccos (- \frac{3}{4}) + 2 πn

4 cos (x) + 3 = 0

将

3

到右边

4 cos (x) + 3 = 0

两边减去

3

4 cos (x) + 3 - 3 = 0 - 3

化简

4 cos (x) = - 3

4 cos (x) = - 3

两边除以

4

4 cos (x) = - 3

两边除以

4

\frac{4 cos ( x )}{4} = \frac{- 3}{4}

化简

cos (x) = - \frac{3}{4}

cos (x) = - \frac{3}{4}

使用反三角函数性质

cos (x) = - \frac{3}{4}

cos (x) = - \frac{3}{4}

的通解

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

x = arccos (- \frac{3}{4}) + 2 πn, x = - arccos (- \frac{3}{4}) + 2 πn

x = arccos (- \frac{3}{4}) + 2 πn, x = - arccos (- \frac{3}{4}) + 2 πn

合并所有解

x = 2 πn, x = π + 2 πn, x = arccos (- \frac{3}{4}) + 2 πn, x = - arccos (- \frac{3}{4}) + 2 πn

以小数形式表示解

x = 2 πn, x = π + 2 πn, x = 2.41885 \dots + 2 πn, x = - 2.41885 \dots + 2 πn

作图

流行的例子

tan(θ)=(-4sqrt(3))/4

tan (θ) = \frac{- 4 3}{4}

cos (x) = \frac{6}{25}

3sin(φ)-cos(2φ)=1

3 sin (φ) - cos (2 φ) = 1

cos(4x)=2cos(2x)-1

cos (4 x) = 2 cos (2 x) - 1

cot^2(θ)+cot(θ)=0

cot^{2} (θ) + cot (θ) = 0