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受欢迎的三角函数 >

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

解答

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

解答

x = 0.46364 \dots + πn

+1

度数

x = 26.56505 \dots^{\circ} + 18 0^{\circ} n

求解步骤

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

两边减去

\frac{1}{2}

sin (x) + sin^{2} (\frac{x}{2}) - \frac{1}{2} = 0

化简

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

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

将项转换为分式:

sin (x) = \frac{sin ( x ) 2}{2}, sin^{2} (\frac{x}{2}) = \frac{sin ^{2} ( \frac{x}{2} ) 2}{2}

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

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

\frac{2 sin ( x ) + 2 sin ^{2} ( \frac{x}{2} ) - 1}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 sin (x) + 2 sin^{2} (\frac{x}{2}) - 1 = 0

使用三角恒等式改写

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

使用倍角公式:

1 - 2 sin^{2} (x) = cos (2 x)

- 1 + 2 sin^{2} (x) = - cos (2 x)

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

乘

2 \cdot \frac{x}{2} : x

2 \cdot \frac{x}{2}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{x \cdot 2}{2}

约分:

2

= x

= 2 sin (x) - cos (x)

在两边除以

cos (x), cos (x) \neq = 0

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

化简

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

使用基本三角恒等式:

\frac{sin ( x )}{cos ( x )} = tan (x)

- 1 + 2 tan (x) = 0

- 1 + 2 tan (x) = 0

将

1

到右边

- 1 + 2 tan (x) = 0

两边加上

1

- 1 + 2 tan (x) + 1 = 0 + 1

化简

2 tan (x) = 1

2 tan (x) = 1

两边除以

2

2 tan (x) = 1

两边除以

2

\frac{2 tan ( x )}{2} = \frac{1}{2}

化简

tan (x) = \frac{1}{2}

tan (x) = \frac{1}{2}

使用反三角函数性质

tan (x) = \frac{1}{2}

tan (x) = \frac{1}{2}

的通解

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{1}{2}) + πn

x = arctan (\frac{1}{2}) + πn

以小数形式表示解

x = 0.46364 \dots + πn

作图

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