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(sin(2x))/(sin(x))=1.5

解答

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

解答

x = 0.72273 \dots + 2 πn, x = 2 π - 0.72273 \dots + 2 πn

+1

度数

x = 41.40962 \dots^{\circ} + 36 0^{\circ} n, x = 318.59037 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

1.5

\frac{sin ( 2 x )}{sin ( x )} - 1.5 = 0

使用三角恒等式改写

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

使用倍角公式:

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

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

约分:

sin (x)

= - 1.5 + 2 cos (x)

- 1.5 + 2 cos (x) = 0

将

1.5

到右边

- 1.5 + 2 cos (x) = 0

两边加上

1.5

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

化简

2 cos (x) = 1.5

2 cos (x) = 1.5

两边除以

2

2 cos (x) = 1.5

两边除以

2

\frac{2 cos ( x )}{2} = \frac{1.5}{2}

化简

cos (x) = 0.75

cos (x) = 0.75

使用反三角函数性质

cos (x) = 0.75

cos (x) = 0.75

的通解

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

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

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

以小数形式表示解

x = 0.72273 \dots + 2 πn, x = 2 π - 0.72273 \dots + 2 πn

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sin^{2} (x) = \frac{( 5 m - 5 )}{2}

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