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受欢迎的三角函数 >

sin(x+10)=cos(x+20)

解答

sin (x + 1 0^{\circ}) = cos (x + 2 0^{\circ})

解答

x = - 36 0^{\circ} n + 3 0^{\circ}, x = - 15 0^{\circ} - 36 0^{\circ} n

+1

弧度

x = \frac{π}{6} - 2 πn, x = - \frac{5 π}{6} - 2 πn

求解步骤

sin (x + 1 0^{\circ}) = cos (x + 2 0^{\circ})

两边减去

cos (x + 2 0^{\circ})

sin (x + 1 0^{\circ}) - cos (x + 2 0^{\circ}) = 0

化简

sin (x + 1 0^{\circ}) - cos (x + 2 0^{\circ}) : sin (\frac{18 x + 18 0 ^{\circ}}{18}) - cos (\frac{9 x + 18 0 ^{\circ}}{9})

sin (x + 1 0^{\circ}) - cos (x + 2 0^{\circ})

化简

x + 1 0^{\circ} : \frac{18 x + 18 0 ^{\circ}}{18}

x + 1 0^{\circ}

将项转换为分式:

x = \frac{x 18}{18}

= \frac{x \cdot 18}{18} + 1 0^{\circ}

因为分母相等，所以合并分式:

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

= \frac{x \cdot 18 + 18 0 ^{\circ}}{18}

= sin (\frac{18 x + 18 0 ^{\circ}}{18}) - cos (x + 2 0^{\circ})

化简

x + 2 0^{\circ} : \frac{9 x + 18 0 ^{\circ}}{9}

x + 2 0^{\circ}

将项转换为分式:

x = \frac{x 9}{9}

= \frac{x \cdot 9}{9} + 2 0^{\circ}

因为分母相等，所以合并分式:

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

= \frac{x \cdot 9 + 18 0 ^{\circ}}{9}

= sin (\frac{18 x + 18 0 ^{\circ}}{18}) - cos (\frac{9 x + 18 0 ^{\circ}}{9})

sin (\frac{18 x + 18 0 ^{\circ}}{18}) - cos (\frac{9 x + 18 0 ^{\circ}}{9}) = 0

使用三角恒等式改写

- cos (\frac{18 0 ^{\circ} + 9 x}{9}) + sin (\frac{18 0 ^{\circ} + 18 x}{18})

利用以下特性:

sin (x) = cos (9 0^{\circ} - x)

= - cos (\frac{18 0 ^{\circ} + 9 x}{9}) + cos (9 0^{\circ} - \frac{18 0 ^{\circ} + 18 x}{18})

化简

9 0^{\circ} - \frac{18 0 ^{\circ} + 18 x}{18} : \frac{72 0 ^{\circ} - 9 x}{9}

9 0^{\circ} - \frac{18 0 ^{\circ} + 18 x}{18}

2, 18

的最小公倍数:

18

2, 18

最小公倍数 (LCM)

2

质因数分解:

2

2

2

是质数，因此无法因数分解

= 2

18

质因数分解:

2 \cdot 3 \cdot 3

18

18

除以

218 = 9 \cdot 2

= 2 \cdot 9

9

除以

39 = 3 \cdot 3

= 2 \cdot 3 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 3 \cdot 3

将每个因子乘以它在

2

或

18

中出现的最多次数

= 2 \cdot 3 \cdot 3

数字相乘:

2 \cdot 3 \cdot 3 = 18

= 18

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

18

对于

9 0^{\circ} :

将分母和分子乘以

9

9 0^{\circ} = \frac{18 0 ^{\circ} 9}{2 \cdot 9} = 9 0^{\circ}

= 9 0^{\circ} - \frac{18 0 ^{\circ} + 18 x}{18}

因为分母相等，所以合并分式:

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

= \frac{18 0 ^{\circ} 9 - ( 18 0 ^{\circ} + 18 x )}{18}

乘开

18 0^{\circ} 9 - (18 0^{\circ} + 18 x) : 144 0^{\circ} - 18 x

18 0^{\circ} 9 - (18 0^{\circ} + 18 x)

= 162 0^{\circ} - (18 0^{\circ} + 18 x)

- (18 0^{\circ} + 18 x) : - 18 0^{\circ} - 18 x

- (18 0^{\circ} + 18 x)

打开括号

= - (18 0^{\circ}) - (18 x)

使用加减运算法则

+ (- a) = - a

= - 18 0^{\circ} - 18 x

= 18 0^{\circ} 9 - 18 0^{\circ} - 18 x

同类项相加:

162 0^{\circ} - 18 0^{\circ} = 144 0^{\circ}

= 144 0^{\circ} - 18 x

= \frac{144 0 ^{\circ} - 18 x}{18}

分解

144 0^{\circ} - 18 x : 2 (72 0^{\circ} - 9 x)

144 0^{\circ} - 18 x

改写为

= 2 \cdot 72 0^{\circ} - 2 \cdot 9 x

因式分解出通项

2

= 2 (72 0^{\circ} - 9 x)

= \frac{2 ( 72 0 ^{\circ} - 9 x )}{18}

约分:

2

= \frac{72 0 ^{\circ} - 9 x}{9}

= - cos (\frac{18 0 ^{\circ} + 9 x}{9}) + cos (\frac{72 0 ^{\circ} - 9 x}{9})

使用和差化积恒等式:

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

= - 2 sin (\frac{\frac{72 0 ^{\circ} - 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9}}{2}) sin (\frac{\frac{72 0 ^{\circ} - 9 x}{9} - \frac{18 0 ^{\circ} + 9 x}{9}}{2})

化简

- 2 sin (\frac{\frac{72 0 ^{\circ} - 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9}}{2}) sin (\frac{\frac{72 0 ^{\circ} - 9 x}{9} - \frac{18 0 ^{\circ} + 9 x}{9}}{2}) : - 2 sin (5 0^{\circ}) sin (\frac{- 6 x + 18 0 ^{\circ}}{6})

- 2 sin (\frac{\frac{72 0 ^{\circ} - 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9}}{2}) sin (\frac{\frac{72 0 ^{\circ} - 9 x}{9} - \frac{18 0 ^{\circ} + 9 x}{9}}{2})

\frac{\frac{72 0 ^{\circ} - 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9}}{2} = 5 0^{\circ}

\frac{\frac{72 0 ^{\circ} - 9 x}{9} + \frac{18 0 ^{\circ} + 9 x}{9}}{2}

合并分式

\frac{- 9 x + 72 0 ^{\circ}}{9} + \frac{9 x + 18 0 ^{\circ}}{9} : 10 0^{\circ}

使用法则

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

= \frac{72 0 ^{\circ} - 9 x + 18 0 ^{\circ} + 9 x}{9}

72 0^{\circ} - 9 x + 18 0^{\circ} + 9 x = 90 0^{\circ}

72 0^{\circ} - 9 x + 18 0^{\circ} + 9 x

对同类项分组

= - 9 x + 9 x + 72 0^{\circ} + 18 0^{\circ}

同类项相加:

- 9 x + 9 x = 0

= 72 0^{\circ} + 18 0^{\circ}

同类项相加:

72 0^{\circ} + 18 0^{\circ} = 90 0^{\circ}

= 90 0^{\circ}

= 90 0^{\circ}

= \frac{10 0 ^{\circ}}{2}

使用分式法则:

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

= \frac{90 0 ^{\circ}}{9 \cdot 2}

数字相乘:

9 \cdot 2 = 18

= 5 0^{\circ}

= - 2 sin (5 0^{\circ}) sin (\frac{\frac{- 9 x + 72 0 ^{\circ}}{9} - \frac{9 x + 18 0 ^{\circ}}{9}}{2})

\frac{\frac{72 0 ^{\circ} - 9 x}{9} - \frac{18 0 ^{\circ} + 9 x}{9}}{2} = \frac{- 6 x + 18 0 ^{\circ}}{6}

\frac{\frac{72 0 ^{\circ} - 9 x}{9} - \frac{18 0 ^{\circ} + 9 x}{9}}{2}

合并分式

\frac{- 9 x + 72 0 ^{\circ}}{9} - \frac{9 x + 18 0 ^{\circ}}{9} : \frac{72 0 ^{\circ} - 9 x - ( 18 0 ^{\circ} + 9 x )}{9}

使用法则

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

= \frac{72 0 ^{\circ} - 9 x - ( 9 x + 18 0 ^{\circ} )}{9}

= \frac{\frac{72 0 ^{\circ} - 9 x - ( 9 x + 18 0 ^{\circ} )}{9}}{2}

使用分式法则:

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

= \frac{72 0 ^{\circ} - 9 x - ( 18 0 ^{\circ} + 9 x )}{9 \cdot 2}

数字相乘:

9 \cdot 2 = 18

= \frac{72 0 ^{\circ} - 9 x - ( 9 x + 18 0 ^{\circ} )}{18}

乘开

72 0^{\circ} - 9 x - (18 0^{\circ} + 9 x) : - 18 x + 54 0^{\circ}

72 0^{\circ} - 9 x - (18 0^{\circ} + 9 x)

- (18 0^{\circ} + 9 x) : - 18 0^{\circ} - 9 x

- (18 0^{\circ} + 9 x)

打开括号

= - (18 0^{\circ}) - (9 x)

使用加减运算法则

+ (- a) = - a

= - 18 0^{\circ} - 9 x

= 72 0^{\circ} - 9 x - 18 0^{\circ} - 9 x

化简

72 0^{\circ} - 9 x - 18 0^{\circ} - 9 x : - 18 x + 54 0^{\circ}

72 0^{\circ} - 9 x - 18 0^{\circ} - 9 x

对同类项分组

= - 9 x - 9 x + 72 0^{\circ} - 18 0^{\circ}

同类项相加:

- 9 x - 9 x = - 18 x

= - 18 x + 72 0^{\circ} - 18 0^{\circ}

同类项相加:

72 0^{\circ} - 18 0^{\circ} = 54 0^{\circ}

= - 18 x + 54 0^{\circ}

= - 18 x + 54 0^{\circ}

= \frac{- 18 x + 54 0 ^{\circ}}{18}

分解

- 18 x + 54 0^{\circ} : 3 (- 6 x + 18 0^{\circ})

- 18 x + 54 0^{\circ}

改写为

= - 3 \cdot 6 x + 54 0^{\circ}

因式分解出通项

3

= 3 (- 6 x + 18 0^{\circ})

= \frac{3 ( - 6 x + 18 0 ^{\circ} )}{18}

约分:

3

= \frac{- 6 x + 18 0 ^{\circ}}{6}

= - 2 sin (5 0^{\circ}) sin (\frac{- 6 x + 18 0 ^{\circ}}{6})

= - 2 sin (5 0^{\circ}) sin (\frac{- 6 x + 18 0 ^{\circ}}{6})

- 2 sin (5 0^{\circ}) sin (\frac{- 6 x + 18 0 ^{\circ}}{6}) = 0

两边除以

- 2 sin (5 0^{\circ})

- 2 sin (5 0^{\circ}) sin (\frac{- 6 x + 18 0 ^{\circ}}{6}) = 0

两边除以

- 2 sin (5 0^{\circ})

\frac{- 2 sin ( 5 0 ^{\circ} ) sin ( \frac{- 6 x + 18 0 ^{\circ}}{6} )}{- 2 sin ( 5 0 ^{\circ} )} = \frac{0}{- 2 sin ( 5 0 ^{\circ} )}

化简

sin (\frac{- 6 x + 18 0 ^{\circ}}{6}) = 0

sin (\frac{- 6 x + 18 0 ^{\circ}}{6}) = 0

sin (\frac{- 6 x + 18 0 ^{\circ}}{6}) = 0

的通解

sin (x)

周期表（周期为

36 0^{\circ} n

"）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{- 6 x + 18 0 ^{\circ}}{6} = 0 + 36 0^{\circ} n, \frac{- 6 x + 18 0 ^{\circ}}{6} = 18 0^{\circ} + 36 0^{\circ} n

\frac{- 6 x + 18 0 ^{\circ}}{6} = 0 + 36 0^{\circ} n, \frac{- 6 x + 18 0 ^{\circ}}{6} = 18 0^{\circ} + 36 0^{\circ} n

解

\frac{- 6 x + 18 0 ^{\circ}}{6} = 0 + 36 0^{\circ} n : x = - 36 0^{\circ} n + 3 0^{\circ}

\frac{- 6 x + 18 0 ^{\circ}}{6} = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

\frac{- 6 x + 18 0 ^{\circ}}{6} = 36 0^{\circ} n

在两边乘以

6

\frac{- 6 x + 18 0 ^{\circ}}{6} = 36 0^{\circ} n

在两边乘以

6

\frac{6 ( - 6 x + 18 0 ^{\circ} )}{6} = 6 \cdot 36 0^{\circ} n

化简

- 6 x + 18 0^{\circ} = 216 0^{\circ} n

- 6 x + 18 0^{\circ} = 216 0^{\circ} n

将

18 0^{\circ}

到右边

- 6 x + 18 0^{\circ} = 216 0^{\circ} n

两边减去

18 0^{\circ}

- 6 x + 18 0^{\circ} - 18 0^{\circ} = 216 0^{\circ} n - 18 0^{\circ}

化简

- 6 x = 216 0^{\circ} n - 18 0^{\circ}

- 6 x = 216 0^{\circ} n - 18 0^{\circ}

两边除以

- 6

- 6 x = 216 0^{\circ} n - 18 0^{\circ}

两边除以

- 6

\frac{- 6 x}{- 6} = \frac{216 0 ^{\circ} n}{- 6} - \frac{18 0 ^{\circ}}{- 6}

化简

\frac{- 6 x}{- 6} = \frac{216 0 ^{\circ} n}{- 6} - \frac{18 0 ^{\circ}}{- 6}

化简

\frac{- 6 x}{- 6} : x

\frac{- 6 x}{- 6}

使用分式法则:

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

= \frac{6 x}{6}

数字相除:

\frac{6}{6} = 1

= x

化简

\frac{216 0 ^{\circ} n}{- 6} - \frac{18 0 ^{\circ}}{- 6} : - 36 0^{\circ} n + 3 0^{\circ}

\frac{216 0 ^{\circ} n}{- 6} - \frac{18 0 ^{\circ}}{- 6}

\frac{216 0 ^{\circ} n}{- 6} = - 36 0^{\circ} n

\frac{216 0 ^{\circ} n}{- 6}

使用分式法则:

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

= - \frac{216 0 ^{\circ} n}{6}

数字相除:

\frac{12}{6} = 2

= - 36 0^{\circ} n

= - 36 0^{\circ} n - \frac{18 0 ^{\circ}}{- 6}

使用分式法则:

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

= - 36 0^{\circ} n - (- 3 0^{\circ})

使用法则

- (- a) = a

= - 36 0^{\circ} n + 3 0^{\circ}

x = - 36 0^{\circ} n + 3 0^{\circ}

x = - 36 0^{\circ} n + 3 0^{\circ}

x = - 36 0^{\circ} n + 3 0^{\circ}

解

\frac{- 6 x + 18 0 ^{\circ}}{6} = 18 0^{\circ} + 36 0^{\circ} n : x = - 15 0^{\circ} - 36 0^{\circ} n

\frac{- 6 x + 18 0 ^{\circ}}{6} = 18 0^{\circ} + 36 0^{\circ} n

在两边乘以

6

\frac{- 6 x + 18 0 ^{\circ}}{6} = 18 0^{\circ} + 36 0^{\circ} n

在两边乘以

6

\frac{6 ( - 6 x + 18 0 ^{\circ} )}{6} = 108 0^{\circ} + 6 \cdot 36 0^{\circ} n

化简

- 6 x + 18 0^{\circ} = 108 0^{\circ} + 216 0^{\circ} n

- 6 x + 18 0^{\circ} = 108 0^{\circ} + 216 0^{\circ} n

将

18 0^{\circ}

到右边

- 6 x + 18 0^{\circ} = 108 0^{\circ} + 216 0^{\circ} n

两边减去

18 0^{\circ}

- 6 x + 18 0^{\circ} - 18 0^{\circ} = 108 0^{\circ} + 216 0^{\circ} n - 18 0^{\circ}

化简

- 6 x = 90 0^{\circ} + 216 0^{\circ} n

- 6 x = 90 0^{\circ} + 216 0^{\circ} n

两边除以

- 6

- 6 x = 90 0^{\circ} + 216 0^{\circ} n

两边除以

- 6

\frac{- 6 x}{- 6} = \frac{90 0 ^{\circ}}{- 6} + \frac{216 0 ^{\circ} n}{- 6}

化简

\frac{- 6 x}{- 6} = \frac{90 0 ^{\circ}}{- 6} + \frac{216 0 ^{\circ} n}{- 6}

化简

\frac{- 6 x}{- 6} : x

\frac{- 6 x}{- 6}

使用分式法则:

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

= \frac{6 x}{6}

数字相除:

\frac{6}{6} = 1

= x

化简

\frac{90 0 ^{\circ}}{- 6} + \frac{216 0 ^{\circ} n}{- 6} : - 15 0^{\circ} - 36 0^{\circ} n

\frac{90 0 ^{\circ}}{- 6} + \frac{216 0 ^{\circ} n}{- 6}

使用分式法则:

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

= - 15 0^{\circ} + \frac{216 0 ^{\circ} n}{- 6}

\frac{216 0 ^{\circ} n}{- 6} = - 36 0^{\circ} n

\frac{216 0 ^{\circ} n}{- 6}

使用分式法则:

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

= - \frac{216 0 ^{\circ} n}{6}

数字相除:

\frac{12}{6} = 2

= - 36 0^{\circ} n

= - 15 0^{\circ} - 36 0^{\circ} n

x = - 15 0^{\circ} - 36 0^{\circ} n

x = - 15 0^{\circ} - 36 0^{\circ} n

x = - 15 0^{\circ} - 36 0^{\circ} n

x = - 36 0^{\circ} n + 3 0^{\circ}, x = - 15 0^{\circ} - 36 0^{\circ} n

作图

流行的例子

sin (b) = \frac{2}{3}

(-5cos(x)-6sin(x))^2-11sin^2(x)=25

(- 5 cos (x) - 6 sin (x))^{2} - 11 sin^{2} (x) = 25

2sin^2(x)-9sin(x)-5=0

2 sin^{2} (x) - 9 sin (x) - 5 = 0

2sin^2(x)=5cos(x)-1

2 sin^{2} (x) = 5 cos (x) - 1

solvefor x,sec(x+10)=5.759

so l v e f or x, sec (x + 1 0^{\circ}) = 5.759