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受欢迎的三角函数 >

sin(x)+cos(x)= 5/7 ,90>x>0

解答

sin (x) + cos (x) = \frac{5}{7}, 9 0^{\circ} > x > 0^{\circ}

解答

x \in R 无解

求解步骤

sin (x) + cos (x) = \frac{5}{7}, 9 0^{\circ} > x > 0^{\circ}

使用三角恒等式改写

sin (x) + cos (x)

sin (x) + cos (x) = 2 sin (x + 4 5^{\circ})

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 (4 5^{\circ}) sin (x) + sin (4 5^{\circ}) cos (x))

使用角和恒等式:

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

= 2 sin (x + 4 5^{\circ})

= 2 sin (x + 4 5^{\circ})

2 sin (x + 4 5^{\circ}) = \frac{5}{7}

两边除以

2

2 sin (x + 4 5^{\circ}) = \frac{5}{7}

两边除以

2

\frac{2 sin ( x + 4 5 ^{\circ} )}{2} = \frac{\frac{5}{7}}{2}

化简

\frac{2 sin ( x + 4 5 ^{\circ} )}{2} = \frac{\frac{5}{7}}{2}

化简

\frac{2 sin ( x + 4 5 ^{\circ} )}{2} : sin (x + 4 5^{\circ})

\frac{2 sin ( x + 4 5 ^{\circ} )}{2}

约分:

2

= sin (x + 4 5^{\circ})

化简

\frac{\frac{5}{7}}{2} : \frac{5 2}{14}

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

使用分式法则:

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

= \frac{5}{7 2}

\frac{5}{7 2}

有理化:

\frac{5 2}{14}

\frac{5}{7 2}

乘以共轭根式

\frac{2}{2}

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

722 = 14

722

使用根式运算法则:

a a = a

22 = 2

= 7 \cdot 2

数字相乘:

7 \cdot 2 = 14

= 14

= \frac{5 2}{14}

= \frac{5 2}{14}

sin (x + 4 5^{\circ}) = \frac{5 2}{14}

sin (x + 4 5^{\circ}) = \frac{5 2}{14}

sin (x + 4 5^{\circ}) = \frac{5 2}{14}

使用反三角函数性质

sin (x + 4 5^{\circ}) = \frac{5 2}{14}

sin (x + 4 5^{\circ}) = \frac{5 2}{14}

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (a) + 36 0^{\circ} n

x + 4 5^{\circ} = arcsin (\frac{5 2}{14}) + 36 0^{\circ} n, x + 4 5^{\circ} = 18 0^{\circ} - arcsin (\frac{5 2}{14}) + 36 0^{\circ} n

x + 4 5^{\circ} = arcsin (\frac{5 2}{14}) + 36 0^{\circ} n, x + 4 5^{\circ} = 18 0^{\circ} - arcsin (\frac{5 2}{14}) + 36 0^{\circ} n

解

x + 4 5^{\circ} = arcsin (\frac{5 2}{14}) + 36 0^{\circ} n : x = arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

x + 4 5^{\circ} = arcsin (\frac{5 2}{14}) + 36 0^{\circ} n

化简

arcsin (\frac{5 2}{14}) + 36 0^{\circ} n : arcsin (\frac{5}{7 2}) + 36 0^{\circ} n

arcsin (\frac{5 2}{14}) + 36 0^{\circ} n

\frac{5 2}{14} = \frac{5}{7 2}

\frac{5 2}{14}

分解

14 : 2 \cdot 7

因式分解

14 = 2 \cdot 7

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

消掉

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

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

使用根式运算法则:

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

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

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

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

数字相减:

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

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

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

= arcsin (\frac{5}{7 2}) + 36 0^{\circ} n

x + 4 5^{\circ} = arcsin (\frac{5}{7 2}) + 36 0^{\circ} n

将

4 5^{\circ}

到右边

x + 4 5^{\circ} = arcsin (\frac{5}{7 2}) + 36 0^{\circ} n

两边减去

4 5^{\circ}

x + 4 5^{\circ} - 4 5^{\circ} = arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

化简

x + 4 5^{\circ} - 4 5^{\circ} = arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

化简

x + 4 5^{\circ} - 4 5^{\circ} : x

x + 4 5^{\circ} - 4 5^{\circ}

同类项相加:

4 5^{\circ} - 4 5^{\circ} = 0

= x

化简

arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ} : arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

= arcsin (\frac{5 2}{14}) + 36 0^{\circ} n - 4 5^{\circ}

\frac{5 2}{14} = \frac{5}{7 2}

\frac{5 2}{14}

分解

14 : 2 \cdot 7

因式分解

14 = 2 \cdot 7

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

消掉

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

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

使用根式运算法则:

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

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

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

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

数字相减:

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

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

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

= arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

无法进一步化简

= arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

x = arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

x = arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

x = arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

解

x + 4 5^{\circ} = 18 0^{\circ} - arcsin (\frac{5 2}{14}) + 36 0^{\circ} n : x = 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

x + 4 5^{\circ} = 18 0^{\circ} - arcsin (\frac{5 2}{14}) + 36 0^{\circ} n

化简

18 0^{\circ} - arcsin (\frac{5 2}{14}) + 36 0^{\circ} n : 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n

18 0^{\circ} - arcsin (\frac{5 2}{14}) + 36 0^{\circ} n

\frac{5 2}{14} = \frac{5}{7 2}

\frac{5 2}{14}

分解

14 : 2 \cdot 7

因式分解

14 = 2 \cdot 7

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

消掉

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

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

使用根式运算法则:

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

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

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

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

数字相减:

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

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

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

= 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n

x + 4 5^{\circ} = 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n

将

4 5^{\circ}

到右边

x + 4 5^{\circ} = 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n

两边减去

4 5^{\circ}

x + 4 5^{\circ} - 4 5^{\circ} = 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

化简

x + 4 5^{\circ} - 4 5^{\circ} = 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

化简

x + 4 5^{\circ} - 4 5^{\circ} : x

x + 4 5^{\circ} - 4 5^{\circ}

同类项相加:

4 5^{\circ} - 4 5^{\circ} = 0

= x

化简

18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ} : 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

= 18 0^{\circ} - arcsin (\frac{5 2}{14}) + 36 0^{\circ} n - 4 5^{\circ}

\frac{5 2}{14} = \frac{5}{7 2}

\frac{5 2}{14}

分解

14 : 2 \cdot 7

因式分解

14 = 2 \cdot 7

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

消掉

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

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

使用根式运算法则:

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

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

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

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

数字相减:

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

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

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

= 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

无法进一步化简

= 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

x = 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

x = 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

x = 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

x = arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}, x = 18 0^{\circ} - arcsin (\frac{5}{7 2}) + 36 0^{\circ} n - 4 5^{\circ}

在

9 0^{\circ} > x > 0

范围内的解

x \in R 无解

作图

流行的例子

2cos^2(θ)+cos(θ)-1=0,\forall 0<= θ<2pi

2 cos^{2} (θ) + cos (θ) - 1 = 0, \forall 0 \leq θ < 2 π

2 cos (4 5^{\circ} - x) = 1

sin (θ) = 0.35

sin (θ) = 0.07

9sin^2(x)tan(x)=16tan(x)

9 sin^{2} (x) tan (x) = 16 tan (x)