zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

sin(θ)+cos(θ)= 7/5 ,sin(2θ)

解答

sin (θ) + cos (θ) = \frac{7}{5}, sin (2 θ)

解答

θ \in R 无解

求解步骤

sin (θ) + cos (θ) = \frac{7}{5}, sin (2 θ)

使用三角恒等式改写

sin (θ) + cos (θ)

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

sin (θ) + cos (θ)

改写为

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

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 (θ) + sin (\frac{π}{4}) cos (θ))

使用角和恒等式:

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

= 2 sin (θ + \frac{π}{4})

= 2 sin (θ + \frac{π}{4})

2 sin (θ + \frac{π}{4}) = \frac{7}{5}

两边除以

2

2 sin (θ + \frac{π}{4}) = \frac{7}{5}

两边除以

2

\frac{2 sin ( θ + \frac{π}{4} )}{2} = \frac{\frac{7}{5}}{2}

化简

\frac{2 sin ( θ + \frac{π}{4} )}{2} = \frac{\frac{7}{5}}{2}

化简

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

\frac{2 sin ( θ + \frac{π}{4} )}{2}

约分:

2

= sin (θ + \frac{π}{4})

化简

\frac{\frac{7}{5}}{2} : \frac{7 2}{10}

\frac{\frac{7}{5}}{2}

使用分式法则:

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

= \frac{7}{5 2}

\frac{7}{5 2}

有理化:

\frac{7 2}{10}

\frac{7}{5 2}

乘以共轭根式

\frac{2}{2}

= \frac{7 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{7 2}{10}

= \frac{7 2}{10}

sin (θ + \frac{π}{4}) = \frac{7 2}{10}

sin (θ + \frac{π}{4}) = \frac{7 2}{10}

sin (θ + \frac{π}{4}) = \frac{7 2}{10}

使用反三角函数性质

sin (θ + \frac{π}{4}) = \frac{7 2}{10}

sin (θ + \frac{π}{4}) = \frac{7 2}{10}

的通解

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

θ + \frac{π}{4} = arcsin (\frac{7 2}{10}) + 2 πn, θ + \frac{π}{4} = π - arcsin (\frac{7 2}{10}) + 2 πn

θ + \frac{π}{4} = arcsin (\frac{7 2}{10}) + 2 πn, θ + \frac{π}{4} = π - arcsin (\frac{7 2}{10}) + 2 πn

解

θ + \frac{π}{4} = arcsin (\frac{7 2}{10}) + 2 πn : θ = arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

θ + \frac{π}{4} = arcsin (\frac{7 2}{10}) + 2 πn

化简

arcsin (\frac{7 2}{10}) + 2 πn : arcsin (\frac{7}{5 2}) + 2 πn

arcsin (\frac{7 2}{10}) + 2 πn

\frac{7 2}{10} = \frac{7}{5 2}

\frac{7 2}{10}

分解

10 : 2 \cdot 5

因式分解

10 = 2 \cdot 5

= \frac{7 2}{2 \cdot 5}

消掉

\frac{7 2}{2 \cdot 5} : \frac{7}{2 \cdot 5}

\frac{7 2}{2 \cdot 5}

使用根式运算法则:

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

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

= \frac{7 \cdot 2 ^{\frac{1}{2}}}{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{7}{5 \cdot 2 ^{- \frac{1}{2} + 1}}

数字相减:

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

= \frac{7}{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{7}{5 2}

= \frac{7}{2 \cdot 5}

= arcsin (\frac{7}{5 2}) + 2 πn

θ + \frac{π}{4} = arcsin (\frac{7}{5 2}) + 2 πn

将

\frac{π}{4}

到右边

θ + \frac{π}{4} = arcsin (\frac{7}{5 2}) + 2 πn

两边减去

\frac{π}{4}

θ + \frac{π}{4} - \frac{π}{4} = arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

化简

θ + \frac{π}{4} - \frac{π}{4} = arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

化简

θ + \frac{π}{4} - \frac{π}{4} : θ

θ + \frac{π}{4} - \frac{π}{4}

同类项相加:

\frac{π}{4} - \frac{π}{4} = 0

= θ

化简

arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4} : arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

= arcsin (\frac{7 2}{10}) + 2 πn - \frac{π}{4}

\frac{7 2}{10} = \frac{7}{5 2}

\frac{7 2}{10}

分解

10 : 2 \cdot 5

因式分解

10 = 2 \cdot 5

= \frac{7 2}{2 \cdot 5}

消掉

\frac{7 2}{2 \cdot 5} : \frac{7}{2 \cdot 5}

\frac{7 2}{2 \cdot 5}

使用根式运算法则:

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

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

= \frac{7 \cdot 2 ^{\frac{1}{2}}}{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{7}{5 \cdot 2 ^{- \frac{1}{2} + 1}}

数字相减:

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

= \frac{7}{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{7}{5 2}

= \frac{7}{2 \cdot 5}

= arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

无法进一步化简

= arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

θ = arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

θ = arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

θ = arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

解

θ + \frac{π}{4} = π - arcsin (\frac{7 2}{10}) + 2 πn : θ = π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

θ + \frac{π}{4} = π - arcsin (\frac{7 2}{10}) + 2 πn

化简

π - arcsin (\frac{7 2}{10}) + 2 πn : π - arcsin (\frac{7}{5 2}) + 2 πn

π - arcsin (\frac{7 2}{10}) + 2 πn

\frac{7 2}{10} = \frac{7}{5 2}

\frac{7 2}{10}

分解

10 : 2 \cdot 5

因式分解

10 = 2 \cdot 5

= \frac{7 2}{2 \cdot 5}

消掉

\frac{7 2}{2 \cdot 5} : \frac{7}{2 \cdot 5}

\frac{7 2}{2 \cdot 5}

使用根式运算法则:

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

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

= \frac{7 \cdot 2 ^{\frac{1}{2}}}{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{7}{5 \cdot 2 ^{- \frac{1}{2} + 1}}

数字相减:

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

= \frac{7}{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{7}{5 2}

= \frac{7}{2 \cdot 5}

= π - arcsin (\frac{7}{5 2}) + 2 πn

θ + \frac{π}{4} = π - arcsin (\frac{7}{5 2}) + 2 πn

将

\frac{π}{4}

到右边

θ + \frac{π}{4} = π - arcsin (\frac{7}{5 2}) + 2 πn

两边减去

\frac{π}{4}

θ + \frac{π}{4} - \frac{π}{4} = π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

化简

θ + \frac{π}{4} - \frac{π}{4} = π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

化简

θ + \frac{π}{4} - \frac{π}{4} : θ

θ + \frac{π}{4} - \frac{π}{4}

同类项相加:

\frac{π}{4} - \frac{π}{4} = 0

= θ

化简

π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4} : π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

= π - arcsin (\frac{7 2}{10}) + 2 πn - \frac{π}{4}

\frac{7 2}{10} = \frac{7}{5 2}

\frac{7 2}{10}

分解

10 : 2 \cdot 5

因式分解

10 = 2 \cdot 5

= \frac{7 2}{2 \cdot 5}

消掉

\frac{7 2}{2 \cdot 5} : \frac{7}{2 \cdot 5}

\frac{7 2}{2 \cdot 5}

使用根式运算法则:

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

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

= \frac{7 \cdot 2 ^{\frac{1}{2}}}{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{7}{5 \cdot 2 ^{- \frac{1}{2} + 1}}

数字相减:

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

= \frac{7}{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{7}{5 2}

= \frac{7}{2 \cdot 5}

= π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

无法进一步化简

= π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

θ = π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

θ = π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

θ = π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

θ = arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}, θ = π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

在

sin (2 θ)

范围内的解

θ \in R 无解

作图

流行的例子

tan (x) = \frac{5}{9}

tan(t)=-(sqrt(11))/5 ,cos(t)>0,sin(t)

tan (t) = - \frac{11}{5}, cos (t) > 0, sin (t)

tan(θ+20)tan(90-3θ)=1

tan (θ + 2 0^{\circ}) tan (9 0^{\circ} - 3 θ) = 1

2csc(x)+7/(cos(x))=0

2 csc (x) + \frac{7}{cos ( x )} = 0

csc (θ) = \frac{13}{6}