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受欢迎的三角函数 >

sin(2x)+1.5cos(x)=0

解答

sin (2 x) + 1.5 cos (x) = 0

解答

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - 0.84806 \dots + 2 πn, x = π + 0.84806 \dots + 2 πn

+1

度数

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = - 48.59037 \dots^{\circ} + 36 0^{\circ} n, x = 228.59037 \dots^{\circ} + 36 0^{\circ} n

求解步骤

sin (2 x) + 1.5 cos (x) = 0

使用三角恒等式改写

sin (2 x) + 1.5 cos (x)

使用倍角公式:

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

= 2 sin (x) cos (x) + 1.5 cos (x)

1.5 cos (x) + 2 cos (x) sin (x) = 0

分解

1.5 cos (x) + 2 cos (x) sin (x) : cos (x) (2 sin (x) + 1.5)

1.5 cos (x) + 2 cos (x) sin (x)

因式分解出通项

cos (x)

= cos (x) (1.5 + 2 sin (x))

cos (x) (2 sin (x) + 1.5) = 0

分别求解每个部分

cos (x) = 0 or 2 sin (x) + 1.5 = 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

2 sin (x) + 1.5 = 0 : x = arcsin (- 0.75) + 2 πn, x = π + arcsin (0.75) + 2 πn

2 sin (x) + 1.5 = 0

将

1.5

到右边

2 sin (x) + 1.5 = 0

两边减去

1.5

2 sin (x) + 1.5 - 1.5 = 0 - 1.5

化简

2 sin (x) = - 1.5

2 sin (x) = - 1.5

两边除以

2

2 sin (x) = - 1.5

两边除以

2

\frac{2 sin ( x )}{2} = \frac{- 1.5}{2}

化简

\frac{2 sin ( x )}{2} = \frac{- 1.5}{2}

化简

\frac{2 sin ( x )}{2} : sin (x)

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

数字相除:

\frac{2}{2} = 1

= sin (x)

化简

\frac{- 1.5}{2} : - 0.75

\frac{- 1.5}{2}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1.5}{2}

数字相除:

\frac{1.5}{2} = 0.75

= - 0.75

sin (x) = - 0.75

sin (x) = - 0.75

sin (x) = - 0.75

使用反三角函数性质

sin (x) = - 0.75

sin (x) = - 0.75

的通解

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

x = arcsin (- 0.75) + 2 πn, x = π + arcsin (0.75) + 2 πn

x = arcsin (- 0.75) + 2 πn, x = π + arcsin (0.75) + 2 πn

合并所有解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arcsin (- 0.75) + 2 πn, x = π + arcsin (0.75) + 2 πn

以小数形式表示解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - 0.84806 \dots + 2 πn, x = π + 0.84806 \dots + 2 πn

作图

流行的例子

sin (w π) = 0

cos (x) = - 8

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

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

sin^2(x)=-(1/2)

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

sin(x/2)-cos(x/2)=0

sin (\frac{x}{2}) - cos (\frac{x}{2}) = 0