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受欢迎的三角函数 >

sin(x)+cos(x)= 2/5

解答

sin (x) + cos (x) = \frac{2}{5}

解答

x = 0.28675 \dots + 2 πn - \frac{π}{4}, x = π - 0.28675 \dots + 2 πn - \frac{π}{4}

+1

度数

x = - 28.57005 \dots^{\circ} + 36 0^{\circ} n, x = 118.57005 \dots^{\circ} + 36 0^{\circ} n

求解步骤

sin (x) + cos (x) = \frac{2}{5}

使用三角恒等式改写

sin (x) + cos (x)

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

sin (x) + cos (x)

改写为

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

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 (x) + sin (\frac{π}{4}) cos (x))

使用角和恒等式:

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

= 2 sin (x + \frac{π}{4})

= 2 sin (x + \frac{π}{4})

2 sin (x + \frac{π}{4}) = \frac{2}{5}

两边除以

2

2 sin (x + \frac{π}{4}) = \frac{2}{5}

两边除以

2

\frac{2 sin ( x + \frac{π}{4} )}{2} = \frac{\frac{2}{5}}{2}

化简

\frac{2 sin ( x + \frac{π}{4} )}{2} = \frac{\frac{2}{5}}{2}

化简

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

\frac{2 sin ( x + \frac{π}{4} )}{2}

约分:

2

= sin (x + \frac{π}{4})

化简

\frac{\frac{2}{5}}{2} : \frac{2}{5}

\frac{\frac{2}{5}}{2}

使用分式法则:

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

= \frac{2}{5 2}

使用根式运算法则:

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

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

= \frac{2}{5 \cdot 2 ^{\frac{1}{2}}}

使用指数法则:

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

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

= \frac{2 ^{1 - \frac{1}{2}}}{5}

数字相减:

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

= \frac{2 ^{\frac{1}{2}}}{5}

使用根式运算法则:

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

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

= \frac{2}{5}

sin (x + \frac{π}{4}) = \frac{2}{5}

sin (x + \frac{π}{4}) = \frac{2}{5}

sin (x + \frac{π}{4}) = \frac{2}{5}

使用反三角函数性质

sin (x + \frac{π}{4}) = \frac{2}{5}

sin (x + \frac{π}{4}) = \frac{2}{5}

的通解

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

x + \frac{π}{4} = arcsin (\frac{2}{5}) + 2 πn, x + \frac{π}{4} = π - arcsin (\frac{2}{5}) + 2 πn

x + \frac{π}{4} = arcsin (\frac{2}{5}) + 2 πn, x + \frac{π}{4} = π - arcsin (\frac{2}{5}) + 2 πn

解

x + \frac{π}{4} = arcsin (\frac{2}{5}) + 2 πn : x = arcsin (\frac{2}{5}) + 2 πn - \frac{π}{4}

x + \frac{π}{4} = arcsin (\frac{2}{5}) + 2 πn

将

\frac{π}{4}

到右边

x + \frac{π}{4} = arcsin (\frac{2}{5}) + 2 πn

两边减去

\frac{π}{4}

x + \frac{π}{4} - \frac{π}{4} = arcsin (\frac{2}{5}) + 2 πn - \frac{π}{4}

化简

x = arcsin (\frac{2}{5}) + 2 πn - \frac{π}{4}

x = arcsin (\frac{2}{5}) + 2 πn - \frac{π}{4}

解

x + \frac{π}{4} = π - arcsin (\frac{2}{5}) + 2 πn : x = π - arcsin (\frac{2}{5}) + 2 πn - \frac{π}{4}

x + \frac{π}{4} = π - arcsin (\frac{2}{5}) + 2 πn

将

\frac{π}{4}

到右边

x + \frac{π}{4} = π - arcsin (\frac{2}{5}) + 2 πn

两边减去

\frac{π}{4}

x + \frac{π}{4} - \frac{π}{4} = π - arcsin (\frac{2}{5}) + 2 πn - \frac{π}{4}

化简

x = π - arcsin (\frac{2}{5}) + 2 πn - \frac{π}{4}

x = π - arcsin (\frac{2}{5}) + 2 πn - \frac{π}{4}

x = arcsin (\frac{2}{5}) + 2 πn - \frac{π}{4}, x = π - arcsin (\frac{2}{5}) + 2 πn - \frac{π}{4}

以小数形式表示解

x = 0.28675 \dots + 2 πn - \frac{π}{4}, x = π - 0.28675 \dots + 2 πn - \frac{π}{4}

作图

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