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受欢迎的三角函数 >

solvefor t,x=5(sin(t)+cos(t))

解答

求解

t, x = 5 (sin (t) + cos (t))

解答

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

求解步骤

x = 5 (sin (t) + cos (t))

交换两边

5 (sin (t) + cos (t)) = x

使用三角恒等式改写

5 (sin (t) + cos (t))

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

sin (t) + cos (t)

改写为

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

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 (t) + sin (\frac{π}{4}) cos (t))

使用角和恒等式:

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

= 2 sin (t + \frac{π}{4})

= 52 sin (t + \frac{π}{4})

52 sin (t + \frac{π}{4}) = x

两边除以

52

52 sin (t + \frac{π}{4}) = x

两边除以

52

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

化简

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

化简

\frac{5 2 sin ( t + \frac{π}{4} )}{5 2} : sin (t + \frac{π}{4})

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

数字相除:

\frac{5}{5} = 1

= \frac{2 sin ( t + \frac{π}{4} )}{2}

约分:

2

= sin (t + \frac{π}{4})

化简

\frac{x}{5 2} : \frac{2 x}{10}

\frac{x}{5 2}

乘以共轭根式

\frac{2}{2}

= \frac{x 2}{5 2 2}

522 = 10

522

使用根式运算法则:

a a = a

22 = 2

= 5 \cdot 2

数字相乘:

5 \cdot 2 = 10

= 10

= \frac{2 x}{10}

sin (t + \frac{π}{4}) = \frac{2 x}{10}

sin (t + \frac{π}{4}) = \frac{2 x}{10}

sin (t + \frac{π}{4}) = \frac{2 x}{10}

使用反三角函数性质

sin (t + \frac{π}{4}) = \frac{2 x}{10}

sin (t + \frac{π}{4}) = \frac{2 x}{10}

的通解

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

t + \frac{π}{4} = arcsin (\frac{2 x}{10}) + 2 πn, t + \frac{π}{4} = π + arcsin (- \frac{2 x}{10}) + 2 πn

t + \frac{π}{4} = arcsin (\frac{2 x}{10}) + 2 πn, t + \frac{π}{4} = π + arcsin (- \frac{2 x}{10}) + 2 πn

解

t + \frac{π}{4} = arcsin (\frac{2 x}{10}) + 2 πn : t = arcsin (\frac{x}{5 2}) + 2 πn - \frac{π}{4}

t + \frac{π}{4} = arcsin (\frac{2 x}{10}) + 2 πn

化简

arcsin (\frac{2 x}{10}) + 2 πn : arcsin (\frac{x}{5 2}) + 2 πn

arcsin (\frac{2 x}{10}) + 2 πn

\frac{2 x}{10} = \frac{x}{5 2}

\frac{2 x}{10}

分解

10 : 2 \cdot 5

因式分解

10 = 2 \cdot 5

= \frac{2 x}{2 \cdot 5}

消掉

\frac{2 x}{2 \cdot 5} : \frac{x}{2 \cdot 5}

\frac{2 x}{2 \cdot 5}

使用根式运算法则:

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

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

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

使用指数法则:

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

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

= \frac{x}{5 \cdot 2 ^{- \frac{1}{2} + 1}}

数字相减:

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

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

使用根式运算法则:

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

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

= \frac{x}{5 2}

= \frac{x}{2 \cdot 5}

= arcsin (\frac{x}{5 2}) + 2 πn

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

将

\frac{π}{4}

到右边

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

两边减去

\frac{π}{4}

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

化简

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

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

解

t + \frac{π}{4} = π + arcsin (- \frac{2 x}{10}) + 2 πn : t = π + arcsin (- \frac{x}{5 2}) + 2 πn - \frac{π}{4}

t + \frac{π}{4} = π + arcsin (- \frac{2 x}{10}) + 2 πn

化简

π + arcsin (- \frac{2 x}{10}) + 2 πn : π + arcsin (- \frac{x}{5 2}) + 2 πn

π + arcsin (- \frac{2 x}{10}) + 2 πn

\frac{2 x}{10} = \frac{x}{5 2}

\frac{2 x}{10}

分解

10 : 2 \cdot 5

因式分解

10 = 2 \cdot 5

= \frac{2 x}{2 \cdot 5}

消掉

\frac{2 x}{2 \cdot 5} : \frac{x}{2 \cdot 5}

\frac{2 x}{2 \cdot 5}

使用根式运算法则:

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

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

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

使用指数法则:

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

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

= \frac{x}{5 \cdot 2 ^{- \frac{1}{2} + 1}}

数字相减:

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

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

使用根式运算法则:

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

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

= \frac{x}{5 2}

= \frac{x}{2 \cdot 5}

= π + arcsin (- \frac{x}{5 2}) + 2 πn

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

将

\frac{π}{4}

到右边

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

两边减去

\frac{π}{4}

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

化简

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

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

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

作图

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