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受欢迎的三角函数 >

sin(2x-10)=cos(x+40)

解答

sin (2 x - 1 0^{\circ}) = cos (x + 4 0^{\circ})

解答

x = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{9}

+1

弧度

x = \frac{π}{9} + \frac{6 π}{9} n

求解步骤

sin (2 x - 1 0^{\circ}) = cos (x + 4 0^{\circ})

使用三角恒等式改写

sin (2 x - 1 0^{\circ}) = cos (x + 4 0^{\circ})

利用以下特性:

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

sin (2 x - 1 0^{\circ}) = sin (9 0^{\circ} - (x + 4 0^{\circ}))

sin (2 x - 1 0^{\circ}) = sin (9 0^{\circ} - (x + 4 0^{\circ}))

使用反三角函数性质

sin (2 x - 1 0^{\circ}) = sin (9 0^{\circ} - (x + 4 0^{\circ}))

sin (x) = sin (y) \Rightarrow x = y + 2 π n, x = π - y + 2 π n

2 x - 1 0^{\circ} = 9 0^{\circ} - (x + 4 0^{\circ}) + 36 0^{\circ} n, 2 x - 1 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (x + 4 0^{\circ})) + 36 0^{\circ} n

2 x - 1 0^{\circ} = 9 0^{\circ} - (x + 4 0^{\circ}) + 36 0^{\circ} n, 2 x - 1 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (x + 4 0^{\circ})) + 36 0^{\circ} n

2 x - 1 0^{\circ} = 9 0^{\circ} - (x + 4 0^{\circ}) + 36 0^{\circ} n : x = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{9}

2 x - 1 0^{\circ} = 9 0^{\circ} - (x + 4 0^{\circ}) + 36 0^{\circ} n

展开

9 0^{\circ} - (x + 4 0^{\circ}) + 36 0^{\circ} n : - x + 36 0^{\circ} n + 5 0^{\circ}

9 0^{\circ} - (x + 4 0^{\circ}) + 36 0^{\circ} n

- (x + 4 0^{\circ}) : - x - 4 0^{\circ}

- (x + 4 0^{\circ})

打开括号

= - (x) - (4 0^{\circ})

使用加减运算法则

+ (- a) = - a

= - x - 4 0^{\circ}

= 9 0^{\circ} - x - 4 0^{\circ} + 36 0^{\circ} n

化简

9 0^{\circ} - x - 4 0^{\circ} + 36 0^{\circ} n : - x + 36 0^{\circ} n + 5 0^{\circ}

9 0^{\circ} - x - 4 0^{\circ} + 36 0^{\circ} n

对同类项分组

= - x + 36 0^{\circ} n + 9 0^{\circ} - 4 0^{\circ}

2, 9

的最小公倍数:

18

2, 9

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

9

质因数分解:

3 \cdot 3

9

9

除以

39 = 3 \cdot 3

= 3 \cdot 3

将每个因子乘以它在

2

或

9

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}

对于

4 0^{\circ} :

将分母和分子乘以

2

4 0^{\circ} = \frac{36 0 ^{\circ} 2}{9 \cdot 2} = 4 0^{\circ}

= 9 0^{\circ} - 4 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{18 0 ^{\circ} 9 - 72 0 ^{\circ}}{18}

同类项相加:

162 0^{\circ} - 72 0^{\circ} = 90 0^{\circ}

= - x + 36 0^{\circ} n + 5 0^{\circ}

= - x + 36 0^{\circ} n + 5 0^{\circ}

2 x - 1 0^{\circ} = - x + 36 0^{\circ} n + 5 0^{\circ}

将

1 0^{\circ}

到右边

2 x - 1 0^{\circ} = - x + 36 0^{\circ} n + 5 0^{\circ}

两边加上

1 0^{\circ}

2 x - 1 0^{\circ} + 1 0^{\circ} = - x + 36 0^{\circ} n + 5 0^{\circ} + 1 0^{\circ}

化简

2 x - 1 0^{\circ} + 1 0^{\circ} = - x + 36 0^{\circ} n + 5 0^{\circ} + 1 0^{\circ}

化简

2 x - 1 0^{\circ} + 1 0^{\circ} : 2 x

2 x - 1 0^{\circ} + 1 0^{\circ}

同类项相加:

- 1 0^{\circ} + 1 0^{\circ} = 0

= 2 x

化简

- x + 36 0^{\circ} n + 5 0^{\circ} + 1 0^{\circ} : - x + 36 0^{\circ} n + 6 0^{\circ}

- x + 36 0^{\circ} n + 5 0^{\circ} + 1 0^{\circ}

合并分式

5 0^{\circ} + 1 0^{\circ} : 6 0^{\circ}

使用法则

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

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

同类项相加:

90 0^{\circ} + 18 0^{\circ} = 108 0^{\circ}

= 6 0^{\circ}

约分:

6

= 6 0^{\circ}

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

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

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

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

将

x

para o lado esquerdo

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

两边加上

x

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

化简

3 x = 36 0^{\circ} n + 6 0^{\circ}

3 x = 36 0^{\circ} n + 6 0^{\circ}

两边除以

3

3 x = 36 0^{\circ} n + 6 0^{\circ}

两边除以

3

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

化简

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

化简

\frac{3 x}{3} : x

\frac{3 x}{3}

数字相除:

\frac{3}{3} = 1

= x

化简

\frac{36 0 ^{\circ} n}{3} + \frac{6 0 ^{\circ}}{3} : \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{9}

\frac{36 0 ^{\circ} n}{3} + \frac{6 0 ^{\circ}}{3}

使用法则

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

= \frac{36 0 ^{\circ} n + 6 0 ^{\circ}}{3}

化简

36 0^{\circ} n + 6 0^{\circ} : \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{3}

36 0^{\circ} n + 6 0^{\circ}

将项转换为分式:

36 0^{\circ} n = \frac{36 0 ^{\circ} n 3}{3}

= \frac{36 0 ^{\circ} n \cdot 3}{3} + 6 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{36 0 ^{\circ} n \cdot 3 + 18 0 ^{\circ}}{3}

数字相乘:

2 \cdot 3 = 6

= \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{3}

= \frac{\frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{3}}{3}

使用分式法则:

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

= \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{3 \cdot 3}

数字相乘:

3 \cdot 3 = 9

= \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{9}

x = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{9}

x = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{9}

x = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{9}

2 x - 1 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (x + 4 0^{\circ})) + 36 0^{\circ} n : x = 18 0^{\circ} + 36 0^{\circ} n - 4 0^{\circ}

2 x - 1 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (x + 4 0^{\circ})) + 36 0^{\circ} n

展开

18 0^{\circ} - (9 0^{\circ} - (x + 4 0^{\circ})) + 36 0^{\circ} n : 18 0^{\circ} + x - 5 0^{\circ} + 36 0^{\circ} n

18 0^{\circ} - (9 0^{\circ} - (x + 4 0^{\circ})) + 36 0^{\circ} n

乘开

9 0^{\circ} - (x + 4 0^{\circ}) : - x + 5 0^{\circ}

9 0^{\circ} - (x + 4 0^{\circ})

- (x + 4 0^{\circ}) : - x - 4 0^{\circ}

- (x + 4 0^{\circ})

打开括号

= - (x) - (4 0^{\circ})

使用加减运算法则

+ (- a) = - a

= - x - 4 0^{\circ}

= 9 0^{\circ} - x - 4 0^{\circ}

化简

9 0^{\circ} - x - 4 0^{\circ} : - x + 5 0^{\circ}

9 0^{\circ} - x - 4 0^{\circ}

对同类项分组

= - x + 9 0^{\circ} - 4 0^{\circ}

2, 9

的最小公倍数:

18

2, 9

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

9

质因数分解:

3 \cdot 3

9

9

除以

39 = 3 \cdot 3

= 3 \cdot 3

将每个因子乘以它在

2

或

9

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}

对于

4 0^{\circ} :

将分母和分子乘以

2

4 0^{\circ} = \frac{36 0 ^{\circ} 2}{9 \cdot 2} = 4 0^{\circ}

= 9 0^{\circ} - 4 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{18 0 ^{\circ} 9 - 72 0 ^{\circ}}{18}

同类项相加:

162 0^{\circ} - 72 0^{\circ} = 90 0^{\circ}

= - x + 5 0^{\circ}

= - x + 5 0^{\circ}

= 18 0^{\circ} - (- x + 5 0^{\circ}) + 36 0^{\circ} n

- (- x + 5 0^{\circ}) : x - 5 0^{\circ}

- (- x + 5 0^{\circ})

打开括号

= - (- x) - (5 0^{\circ})

使用加减运算法则

- (- a) = a, - (a) = - a

= x - 5 0^{\circ}

= 18 0^{\circ} + x - 5 0^{\circ} + 36 0^{\circ} n

2 x - 1 0^{\circ} = 18 0^{\circ} + x - 5 0^{\circ} + 36 0^{\circ} n

将

1 0^{\circ}

到右边

2 x - 1 0^{\circ} = 18 0^{\circ} + x - 5 0^{\circ} + 36 0^{\circ} n

两边加上

1 0^{\circ}

2 x - 1 0^{\circ} + 1 0^{\circ} = 18 0^{\circ} + x - 5 0^{\circ} + 36 0^{\circ} n + 1 0^{\circ}

化简

2 x - 1 0^{\circ} + 1 0^{\circ} = 18 0^{\circ} + x - 5 0^{\circ} + 36 0^{\circ} n + 1 0^{\circ}

化简

2 x - 1 0^{\circ} + 1 0^{\circ} : 2 x

2 x - 1 0^{\circ} + 1 0^{\circ}

同类项相加:

- 1 0^{\circ} + 1 0^{\circ} = 0

= 2 x

化简

18 0^{\circ} + x - 5 0^{\circ} + 36 0^{\circ} n + 1 0^{\circ} : x + 18 0^{\circ} + 36 0^{\circ} n - 4 0^{\circ}

18 0^{\circ} + x - 5 0^{\circ} + 36 0^{\circ} n + 1 0^{\circ}

对同类项分组

= x + 18 0^{\circ} + 36 0^{\circ} n + 1 0^{\circ} - 5 0^{\circ}

合并分式

1 0^{\circ} - 5 0^{\circ} : - 4 0^{\circ}

使用法则

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

= \frac{18 0 ^{\circ} - 90 0 ^{\circ}}{18}

同类项相加:

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

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

使用分式法则:

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

= - 4 0^{\circ}

约分:

2

= - 4 0^{\circ}

= x + 18 0^{\circ} + 36 0^{\circ} n - 4 0^{\circ}

2 x = x + 18 0^{\circ} + 36 0^{\circ} n - 4 0^{\circ}

2 x = x + 18 0^{\circ} + 36 0^{\circ} n - 4 0^{\circ}

2 x = x + 18 0^{\circ} + 36 0^{\circ} n - 4 0^{\circ}

将

x

para o lado esquerdo

2 x = x + 18 0^{\circ} + 36 0^{\circ} n - 4 0^{\circ}

两边减去

x

2 x - x = x + 18 0^{\circ} + 36 0^{\circ} n - 4 0^{\circ} - x

化简

x = 18 0^{\circ} + 36 0^{\circ} n - 4 0^{\circ}

x = 18 0^{\circ} + 36 0^{\circ} n - 4 0^{\circ}

合并重叠的区间

x = \frac{108 0 ^{\circ} n + 18 0 ^{\circ}}{9}

作图

流行的例子

2 sec^{2} (x) - 8 = 0

sin(x)-2cos(2x)=-1/2

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

4 cos^{4} (x) - 1 = 0

arccos(2x)-arccos(x)= pi/3

arccos (2 x) - arccos (x) = \frac{π}{3}

cos(2θ)= 1/2 ,0<= θ<= 2pi

cos (2 θ) = \frac{1}{2}, 0 \leq θ \leq 2 π