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受欢迎的三角函数 >

sinh(pi/4)

解答

sinh (\frac{π}{4})

解答

\frac{e ^{\frac{π}{2}} - 1}{2 e ^{\frac{π}{4}}}

+1

十进制

0.86867 \dots

求解步骤

sinh (\frac{π}{4})

使用双曲函数恒等式:

sinh (x) = \frac{e ^{x} - e ^{- x}}{2}

= \frac{e ^{\frac{π}{4}} - e ^{- \frac{π}{4}}}{2}

\frac{e ^{\frac{π}{4}} - e ^{- \frac{π}{4}}}{2} = \frac{e ^{\frac{π}{2}} - 1}{2 e ^{\frac{π}{4}}}

\frac{e ^{\frac{π}{4}} - e ^{- \frac{π}{4}}}{2}

使用指数法则:

a^{- b} = \frac{1}{a ^{b}}

= \frac{e ^{\frac{π}{4}} - \frac{1}{e ^{\frac{π}{4}}}}{2}

化简

e^{\frac{π}{4}} - \frac{1}{e ^{\frac{π}{4}}} : \frac{e ^{\frac{π}{2}} - 1}{e ^{\frac{π}{4}}}

e^{\frac{π}{4}} - \frac{1}{e ^{\frac{π}{4}}}

将项转换为分式:

e^{\frac{π}{4}} = \frac{e ^{\frac{π}{4}} e ^{\frac{π}{4}}}{e ^{\frac{π}{4}}}

= \frac{e ^{\frac{π}{4}} e ^{\frac{π}{4}}}{e ^{\frac{π}{4}}} - \frac{1}{e ^{\frac{π}{4}}}

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{e ^{\frac{π}{4}} e ^{\frac{π}{4}} - 1}{e ^{\frac{π}{4}}}

e^{\frac{π}{4}} e^{\frac{π}{4}} - 1 = e^{2 \cdot \frac{π}{4}} - 1

e^{\frac{π}{4}} e^{\frac{π}{4}} - 1

e^{\frac{π}{4}} e^{\frac{π}{4}} = e^{2 \cdot \frac{π}{4}}

e^{\frac{π}{4}} e^{\frac{π}{4}}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

e^{\frac{π}{4}} e^{\frac{π}{4}} = e^{\frac{π}{4} + \frac{π}{4}}

= e^{\frac{π}{4} + \frac{π}{4}}

同类项相加:

\frac{π}{4} + \frac{π}{4} = 2 \cdot \frac{π}{4}

= e^{2 \cdot \frac{π}{4}}

= e^{2 \cdot \frac{π}{4}} - 1

= \frac{e ^{2 \cdot \frac{π}{4}} - 1}{e ^{\frac{π}{4}}}

乘

2 \cdot \frac{π}{4} : \frac{π}{2}

2 \cdot \frac{π}{4}

分式相乘:

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

= \frac{π 2}{4}

约分:

2

= \frac{π}{2}

= \frac{e ^{\frac{π}{2}} - 1}{e ^{\frac{π}{4}}}

= \frac{\frac{e ^{\frac{π}{2}} - 1}{e ^{\frac{π}{4}}}}{2}

使用分式法则:

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

= \frac{e ^{\frac{π}{2}} - 1}{e ^{\frac{π}{4}} \cdot 2}

= \frac{e ^{\frac{π}{2}} - 1}{2 e ^{\frac{π}{4}}}

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