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受欢迎的三角函数 >

sin(α)+cos(α)= 1/2

解答

sin (α) + cos (α) = \frac{1}{2}

解答

α = 0.36136 \dots + 2 πn - \frac{π}{4}, α = π - 0.36136 \dots + 2 πn - \frac{π}{4}

+1

度数

α = - 24.29518 \dots^{\circ} + 36 0^{\circ} n, α = 114.29518 \dots^{\circ} + 36 0^{\circ} n

求解步骤

sin (α) + cos (α) = \frac{1}{2}

使用三角恒等式改写

sin (α) + cos (α)

sin (α) + cos (α) = 2 sin (α + \frac{π}{4})

sin (α) + cos (α)

改写为

= 2 (\frac{1}{2} sin (α) + \frac{1}{2} cos (α))

使用以下普通恒等式:

cos (\frac{π}{4}) = \frac{1}{2}

使用以下普通恒等式:

sin (\frac{π}{4}) = \frac{1}{2}

= 2 (cos (\frac{π}{4}) sin (α) + sin (\frac{π}{4}) cos (α))

使用角和恒等式:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= 2 sin (α + \frac{π}{4})

= 2 sin (α + \frac{π}{4})

2 sin (α + \frac{π}{4}) = \frac{1}{2}

两边除以

2

2 sin (α + \frac{π}{4}) = \frac{1}{2}

两边除以

2

\frac{2 sin ( α + \frac{π}{4} )}{2} = \frac{\frac{1}{2}}{2}

化简

\frac{2 sin ( α + \frac{π}{4} )}{2} = \frac{\frac{1}{2}}{2}

化简

\frac{2 sin ( α + \frac{π}{4} )}{2} : sin (α + \frac{π}{4})

\frac{2 sin ( α + \frac{π}{4} )}{2}

约分:

2

= sin (α + \frac{π}{4})

化简

\frac{\frac{1}{2}}{2} : \frac{2}{4}

\frac{\frac{1}{2}}{2}

使用分式法则:

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

= \frac{1}{2 2}

\frac{1}{2 2}

有理化:

\frac{2}{4}

\frac{1}{2 2}

乘以共轭根式

\frac{2}{2}

= \frac{1 \cdot 2}{2 2 2}

1 \cdot 2 = 2

222 = 4

222

使用指数法则:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

同类项相加:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

分式相乘:

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

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 2^{1 + 1}

数字相加:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2}{4}

= \frac{2}{4}

sin (α + \frac{π}{4}) = \frac{2}{4}

sin (α + \frac{π}{4}) = \frac{2}{4}

sin (α + \frac{π}{4}) = \frac{2}{4}

使用反三角函数性质

sin (α + \frac{π}{4}) = \frac{2}{4}

sin (α + \frac{π}{4}) = \frac{2}{4}

的通解

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

α + \frac{π}{4} = arcsin (\frac{2}{4}) + 2 πn, α + \frac{π}{4} = π - arcsin (\frac{2}{4}) + 2 πn

α + \frac{π}{4} = arcsin (\frac{2}{4}) + 2 πn, α + \frac{π}{4} = π - arcsin (\frac{2}{4}) + 2 πn

解

α + \frac{π}{4} = arcsin (\frac{2}{4}) + 2 πn : α = arcsin (\frac{1}{2 2}) + 2 πn - \frac{π}{4}

α + \frac{π}{4} = arcsin (\frac{2}{4}) + 2 πn

化简

arcsin (\frac{2}{4}) + 2 πn : arcsin (\frac{1}{2 2}) + 2 πn

arcsin (\frac{2}{4}) + 2 πn

\frac{2}{4} = \frac{1}{2 2}

\frac{2}{4}

分解

4 : 2^{2}

因式分解

4 = 2^{2}

= \frac{2}{2 ^{2}}

消掉

\frac{2}{2 ^{2}} : \frac{1}{2 ^{\frac{3}{2}}}

\frac{2}{2 ^{2}}

使用根式运算法则:

n a = a^{\frac{1}{n}}

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}}}{2 ^{2}}

使用指数法则:

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

\frac{2 ^{\frac{1}{2}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{1}{2}}}

= \frac{1}{2 ^{2 - \frac{1}{2}}}

数字相减:

2 - \frac{1}{2} = \frac{3}{2}

= \frac{1}{2 ^{\frac{3}{2}}}

= \frac{1}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

使用指数法则:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{1}{2}}

整理后得

= 22

= \frac{1}{2 2}

= arcsin (\frac{1}{2 2}) + 2 πn

α + \frac{π}{4} = arcsin (\frac{1}{2 2}) + 2 πn

将

\frac{π}{4}

到右边

α + \frac{π}{4} = arcsin (\frac{1}{2 2}) + 2 πn

两边减去

\frac{π}{4}

α + \frac{π}{4} - \frac{π}{4} = arcsin (\frac{1}{2 2}) + 2 πn - \frac{π}{4}

化简

α = arcsin (\frac{1}{2 2}) + 2 πn - \frac{π}{4}

α = arcsin (\frac{1}{2 2}) + 2 πn - \frac{π}{4}

解

α + \frac{π}{4} = π - arcsin (\frac{2}{4}) + 2 πn : α = π - arcsin (\frac{1}{2 2}) + 2 πn - \frac{π}{4}

α + \frac{π}{4} = π - arcsin (\frac{2}{4}) + 2 πn

化简

π - arcsin (\frac{2}{4}) + 2 πn : π - arcsin (\frac{1}{2 2}) + 2 πn

π - arcsin (\frac{2}{4}) + 2 πn

\frac{2}{4} = \frac{1}{2 2}

\frac{2}{4}

分解

4 : 2^{2}

因式分解

4 = 2^{2}

= \frac{2}{2 ^{2}}

消掉

\frac{2}{2 ^{2}} : \frac{1}{2 ^{\frac{3}{2}}}

\frac{2}{2 ^{2}}

使用根式运算法则:

n a = a^{\frac{1}{n}}

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}}}{2 ^{2}}

使用指数法则:

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

\frac{2 ^{\frac{1}{2}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{1}{2}}}

= \frac{1}{2 ^{2 - \frac{1}{2}}}

数字相减:

2 - \frac{1}{2} = \frac{3}{2}

= \frac{1}{2 ^{\frac{3}{2}}}

= \frac{1}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

使用指数法则:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{1}{2}}

整理后得

= 22

= \frac{1}{2 2}

= π - arcsin (\frac{1}{2 2}) + 2 πn

α + \frac{π}{4} = π - arcsin (\frac{1}{2 2}) + 2 πn

将

\frac{π}{4}

到右边

α + \frac{π}{4} = π - arcsin (\frac{1}{2 2}) + 2 πn

两边减去

\frac{π}{4}

α + \frac{π}{4} - \frac{π}{4} = π - arcsin (\frac{1}{2 2}) + 2 πn - \frac{π}{4}

化简

α = π - arcsin (\frac{1}{2 2}) + 2 πn - \frac{π}{4}

α = π - arcsin (\frac{1}{2 2}) + 2 πn - \frac{π}{4}

α = arcsin (\frac{1}{2 2}) + 2 πn - \frac{π}{4}, α = π - arcsin (\frac{1}{2 2}) + 2 πn - \frac{π}{4}

以小数形式表示解

α = 0.36136 \dots + 2 πn - \frac{π}{4}, α = π - 0.36136 \dots + 2 πn - \frac{π}{4}

作图

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