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受欢迎的三角函数 >

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

解答

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

解答

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.38439 \dots + 2 πn, x = π - 0.38439 \dots + 2 πn

+1

度数

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 22.02431 \dots^{\circ} + 36 0^{\circ} n, x = 157.97568 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

3 cos (x)

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

使用三角恒等式改写

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

使用倍角公式:

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

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

化简

= - 3 cos (x) + 8 sin (x) cos (x)

- 3 cos (x) + 8 cos (x) sin (x) = 0

分解

- 3 cos (x) + 8 cos (x) sin (x) : cos (x) (8 sin (x) - 3)

- 3 cos (x) + 8 cos (x) sin (x)

因式分解出通项

cos (x)

= cos (x) (- 3 + 8 sin (x))

cos (x) (8 sin (x) - 3) = 0

分别求解每个部分

cos (x) = 0 or 8 sin (x) - 3 = 0

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

cos (x) = 0

cos (x) = 0

的通解

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

8 sin (x) - 3 = 0 : x = arcsin (\frac{3}{8}) + 2 πn, x = π - arcsin (\frac{3}{8}) + 2 πn

8 sin (x) - 3 = 0

将

3

到右边

8 sin (x) - 3 = 0

两边加上

3

8 sin (x) - 3 + 3 = 0 + 3

化简

8 sin (x) = 3

8 sin (x) = 3

两边除以

8

8 sin (x) = 3

两边除以

8

\frac{8 sin ( x )}{8} = \frac{3}{8}

化简

sin (x) = \frac{3}{8}

sin (x) = \frac{3}{8}

使用反三角函数性质

sin (x) = \frac{3}{8}

sin (x) = \frac{3}{8}

的通解

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

x = arcsin (\frac{3}{8}) + 2 πn, x = π - arcsin (\frac{3}{8}) + 2 πn

x = arcsin (\frac{3}{8}) + 2 πn, x = π - arcsin (\frac{3}{8}) + 2 πn

合并所有解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arcsin (\frac{3}{8}) + 2 πn, x = π - arcsin (\frac{3}{8}) + 2 πn

以小数形式表示解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.38439 \dots + 2 πn, x = π - 0.38439 \dots + 2 πn

作图

流行的例子

sin(b)=((250km)/(590km))sin(23),82

solvefor y,sin(x^2y^2)=x