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受欢迎的三角函数 >

sin(x+20)=cos(2x+25)

解答

sin (x + 2 0^{\circ}) = cos (2 x + 2 5^{\circ})

解答

x = \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{12}, x = - \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

+1

弧度

x = \frac{π}{12} + \frac{8 π}{12} n, x = - \frac{19 π}{36} - \frac{72 π}{36} n

求解步骤

sin (x + 2 0^{\circ}) = cos (2 x + 2 5^{\circ})

使用三角恒等式改写

sin (x + 2 0^{\circ}) = cos (2 x + 2 5^{\circ})

利用以下特性:

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

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

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

使用反三角函数性质

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

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

x + 2 0^{\circ} = 9 0^{\circ} - (2 x + 2 5^{\circ}) + 36 0^{\circ} n, x + 2 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (2 x + 2 5^{\circ})) + 36 0^{\circ} n

x + 2 0^{\circ} = 9 0^{\circ} - (2 x + 2 5^{\circ}) + 36 0^{\circ} n, x + 2 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (2 x + 2 5^{\circ})) + 36 0^{\circ} n

x + 2 0^{\circ} = 9 0^{\circ} - (2 x + 2 5^{\circ}) + 36 0^{\circ} n : x = \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{12}

x + 2 0^{\circ} = 9 0^{\circ} - (2 x + 2 5^{\circ}) + 36 0^{\circ} n

展开

9 0^{\circ} - (2 x + 2 5^{\circ}) + 36 0^{\circ} n : - 2 x + 36 0^{\circ} n + 6 5^{\circ}

9 0^{\circ} - (2 x + 2 5^{\circ}) + 36 0^{\circ} n

- (2 x + 2 5^{\circ}) : - 2 x - 2 5^{\circ}

- (2 x + 2 5^{\circ})

打开括号

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

使用加减运算法则

+ (- a) = - a

= - 2 x - 2 5^{\circ}

= 9 0^{\circ} - 2 x - 2 5^{\circ} + 36 0^{\circ} n

化简

9 0^{\circ} - 2 x - 2 5^{\circ} + 36 0^{\circ} n : - 2 x + 36 0^{\circ} n + 6 5^{\circ}

9 0^{\circ} - 2 x - 2 5^{\circ} + 36 0^{\circ} n

对同类项分组

= - 2 x + 36 0^{\circ} n + 9 0^{\circ} - 2 5^{\circ}

2, 36

的最小公倍数:

36

2, 36

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

36

质因数分解:

2 \cdot 2 \cdot 3 \cdot 3

36

36

除以

236 = 18 \cdot 2

= 2 \cdot 18

18

除以

218 = 9 \cdot 2

= 2 \cdot 2 \cdot 9

9

除以

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 3 \cdot 3

将每个因子乘以它在

2

或

36

中出现的最多次数

= 2 \cdot 2 \cdot 3 \cdot 3

数字相乘:

2 \cdot 2 \cdot 3 \cdot 3 = 36

= 36

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

36

对于

9 0^{\circ} :

将分母和分子乘以

18

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

= 9 0^{\circ} - 2 5^{\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} 18 - 90 0 ^{\circ}}{36}

同类项相加:

324 0^{\circ} - 90 0^{\circ} = 234 0^{\circ}

= - 2 x + 36 0^{\circ} n + 6 5^{\circ}

= - 2 x + 36 0^{\circ} n + 6 5^{\circ}

x + 2 0^{\circ} = - 2 x + 36 0^{\circ} n + 6 5^{\circ}

将

2 0^{\circ}

到右边

x + 2 0^{\circ} = - 2 x + 36 0^{\circ} n + 6 5^{\circ}

两边减去

2 0^{\circ}

x + 2 0^{\circ} - 2 0^{\circ} = - 2 x + 36 0^{\circ} n + 6 5^{\circ} - 2 0^{\circ}

化简

x + 2 0^{\circ} - 2 0^{\circ} = - 2 x + 36 0^{\circ} n + 6 5^{\circ} - 2 0^{\circ}

化简

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

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

同类项相加:

2 0^{\circ} - 2 0^{\circ} = 0

= x

化简

- 2 x + 36 0^{\circ} n + 6 5^{\circ} - 2 0^{\circ} : - 2 x + 36 0^{\circ} n + 4 5^{\circ}

- 2 x + 36 0^{\circ} n + 6 5^{\circ} - 2 0^{\circ}

36, 9

的最小公倍数:

36

36, 9

最小公倍数 (LCM)

36

质因数分解:

2 \cdot 2 \cdot 3 \cdot 3

36

36

除以

236 = 18 \cdot 2

= 2 \cdot 18

18

除以

218 = 9 \cdot 2

= 2 \cdot 2 \cdot 9

9

除以

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 3 \cdot 3

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

36

或

9

中出现的最多次数

= 2 \cdot 2 \cdot 3 \cdot 3

数字相乘:

2 \cdot 2 \cdot 3 \cdot 3 = 36

= 36

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

36

对于

2 0^{\circ} :

将分母和分子乘以

4

2 0^{\circ} = \frac{18 0 ^{\circ} 4}{9 \cdot 4} = 2 0^{\circ}

= 6 5^{\circ} - 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{234 0 ^{\circ} - 18 0 ^{\circ} 4}{36}

同类项相加:

234 0^{\circ} - 72 0^{\circ} = 162 0^{\circ}

= 4 5^{\circ}

约分:

9

= - 2 x + 36 0^{\circ} n + 4 5^{\circ}

x = - 2 x + 36 0^{\circ} n + 4 5^{\circ}

x = - 2 x + 36 0^{\circ} n + 4 5^{\circ}

x = - 2 x + 36 0^{\circ} n + 4 5^{\circ}

将

2 x

para o lado esquerdo

x = - 2 x + 36 0^{\circ} n + 4 5^{\circ}

两边加上

2 x

x + 2 x = - 2 x + 36 0^{\circ} n + 4 5^{\circ} + 2 x

化简

3 x = 36 0^{\circ} n + 4 5^{\circ}

3 x = 36 0^{\circ} n + 4 5^{\circ}

两边除以

3

3 x = 36 0^{\circ} n + 4 5^{\circ}

两边除以

3

\frac{3 x}{3} = \frac{36 0 ^{\circ} n}{3} + \frac{4 5 ^{\circ}}{3}

化简

\frac{3 x}{3} = \frac{36 0 ^{\circ} n}{3} + \frac{4 5 ^{\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{4 5 ^{\circ}}{3} : \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{12}

\frac{36 0 ^{\circ} n}{3} + \frac{4 5 ^{\circ}}{3}

使用法则

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

= \frac{36 0 ^{\circ} n + 4 5 ^{\circ}}{3}

化简

36 0^{\circ} n + 4 5^{\circ} : \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{4}

36 0^{\circ} n + 4 5^{\circ}

将项转换为分式:

36 0^{\circ} n = \frac{36 0 ^{\circ} n 4}{4}

= \frac{36 0 ^{\circ} n \cdot 4}{4} + 4 5^{\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 4 + 18 0 ^{\circ}}{4}

数字相乘:

2 \cdot 4 = 8

= \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{4}

= \frac{\frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{4}}{3}

使用分式法则:

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

= \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{4 \cdot 3}

数字相乘:

4 \cdot 3 = 12

= \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{12}

x = \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{12}

x = \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{12}

x = \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{12}

x + 2 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (2 x + 2 5^{\circ})) + 36 0^{\circ} n : x = - \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

x + 2 0^{\circ} = 18 0^{\circ} - (9 0^{\circ} - (2 x + 2 5^{\circ})) + 36 0^{\circ} n

展开

18 0^{\circ} - (9 0^{\circ} - (2 x + 2 5^{\circ})) + 36 0^{\circ} n : 18 0^{\circ} + 2 x - 6 5^{\circ} + 36 0^{\circ} n

18 0^{\circ} - (9 0^{\circ} - (2 x + 2 5^{\circ})) + 36 0^{\circ} n

乘开

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

9 0^{\circ} - (2 x + 2 5^{\circ})

- (2 x + 2 5^{\circ}) : - 2 x - 2 5^{\circ}

- (2 x + 2 5^{\circ})

打开括号

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

使用加减运算法则

+ (- a) = - a

= - 2 x - 2 5^{\circ}

= 9 0^{\circ} - 2 x - 2 5^{\circ}

化简

9 0^{\circ} - 2 x - 2 5^{\circ} : - 2 x + 6 5^{\circ}

9 0^{\circ} - 2 x - 2 5^{\circ}

对同类项分组

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

2, 36

的最小公倍数:

36

2, 36

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

36

质因数分解:

2 \cdot 2 \cdot 3 \cdot 3

36

36

除以

236 = 18 \cdot 2

= 2 \cdot 18

18

除以

218 = 9 \cdot 2

= 2 \cdot 2 \cdot 9

9

除以

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 3 \cdot 3

将每个因子乘以它在

2

或

36

中出现的最多次数

= 2 \cdot 2 \cdot 3 \cdot 3

数字相乘:

2 \cdot 2 \cdot 3 \cdot 3 = 36

= 36

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

36

对于

9 0^{\circ} :

将分母和分子乘以

18

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

= 9 0^{\circ} - 2 5^{\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} 18 - 90 0 ^{\circ}}{36}

同类项相加:

324 0^{\circ} - 90 0^{\circ} = 234 0^{\circ}

= - 2 x + 6 5^{\circ}

= - 2 x + 6 5^{\circ}

= 18 0^{\circ} - (- 2 x + 6 5^{\circ}) + 36 0^{\circ} n

- (- 2 x + 6 5^{\circ}) : 2 x - 6 5^{\circ}

- (- 2 x + 6 5^{\circ})

打开括号

= - (- 2 x) - (6 5^{\circ})

使用加减运算法则

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

= 2 x - 6 5^{\circ}

= 18 0^{\circ} + 2 x - 6 5^{\circ} + 36 0^{\circ} n

x + 2 0^{\circ} = 18 0^{\circ} + 2 x - 6 5^{\circ} + 36 0^{\circ} n

将

2 0^{\circ}

到右边

x + 2 0^{\circ} = 18 0^{\circ} + 2 x - 6 5^{\circ} + 36 0^{\circ} n

两边减去

2 0^{\circ}

x + 2 0^{\circ} - 2 0^{\circ} = 18 0^{\circ} + 2 x - 6 5^{\circ} + 36 0^{\circ} n - 2 0^{\circ}

化简

x + 2 0^{\circ} - 2 0^{\circ} = 18 0^{\circ} + 2 x - 6 5^{\circ} + 36 0^{\circ} n - 2 0^{\circ}

化简

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

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

同类项相加:

2 0^{\circ} - 2 0^{\circ} = 0

= x

化简

18 0^{\circ} + 2 x - 6 5^{\circ} + 36 0^{\circ} n - 2 0^{\circ} : 2 x + 18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ}

18 0^{\circ} + 2 x - 6 5^{\circ} + 36 0^{\circ} n - 2 0^{\circ}

对同类项分组

= 2 x + 18 0^{\circ} + 36 0^{\circ} n - 2 0^{\circ} - 6 5^{\circ}

9, 36

的最小公倍数:

36

9, 36

最小公倍数 (LCM)

9

质因数分解:

3 \cdot 3

9

9

除以

39 = 3 \cdot 3

= 3 \cdot 3

36

质因数分解:

2 \cdot 2 \cdot 3 \cdot 3

36

36

除以

236 = 18 \cdot 2

= 2 \cdot 18

18

除以

218 = 9 \cdot 2

= 2 \cdot 2 \cdot 9

9

除以

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 3 \cdot 3

将每个因子乘以它在

9

或

36

中出现的最多次数

= 3 \cdot 3 \cdot 2 \cdot 2

数字相乘:

3 \cdot 3 \cdot 2 \cdot 2 = 36

= 36

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

36

对于

2 0^{\circ} :

将分母和分子乘以

4

2 0^{\circ} = \frac{18 0 ^{\circ} 4}{9 \cdot 4} = 2 0^{\circ}

= - 2 0^{\circ} - 6 5^{\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} 4 - 234 0 ^{\circ}}{36}

同类项相加:

- 72 0^{\circ} - 234 0^{\circ} = - 306 0^{\circ}

= \frac{- 306 0 ^{\circ}}{36}

使用分式法则:

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

= 2 x + 18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ}

x = 2 x + 18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ}

x = 2 x + 18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ}

x = 2 x + 18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ}

将

2 x

para o lado esquerdo

x = 2 x + 18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ}

两边减去

2 x

x - 2 x = 2 x + 18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ} - 2 x

化简

- x = 18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ}

- x = 18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ}

两边除以

- 1

- x = 18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ}

两边除以

- 1

\frac{- x}{- 1} = \frac{18 0 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{8 5 ^{\circ}}{- 1}

化简

\frac{- x}{- 1} = \frac{18 0 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{8 5 ^{\circ}}{- 1}

化简

\frac{- x}{- 1} : x

\frac{- x}{- 1}

使用分式法则:

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

= \frac{x}{1}

使用法则

\frac{a}{1} = a

= x

化简

\frac{18 0 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{8 5 ^{\circ}}{- 1} : - \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

\frac{18 0 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{8 5 ^{\circ}}{- 1}

使用法则

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

= \frac{18 0 ^{\circ} + 36 0 ^{\circ} n - 8 5 ^{\circ}}{- 1}

使用分式法则:

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

= - \frac{18 0 ^{\circ} + 36 0 ^{\circ} n - 8 5 ^{\circ}}{1}

化简

18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ} : \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

18 0^{\circ} + 36 0^{\circ} n - 8 5^{\circ}

将项转换为分式:

18 0^{\circ} = 18 0^{\circ}, 36 0^{\circ} n = \frac{36 0 ^{\circ} n 36}{36}

= 18 0^{\circ} + \frac{36 0 ^{\circ} n \cdot 36}{36} - 8 5^{\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} 36 + 36 0 ^{\circ} n \cdot 36 - 306 0 ^{\circ}}{36}

18 0^{\circ} 36 + 36 0^{\circ} n \cdot 36 - 306 0^{\circ} = 342 0^{\circ} + 1296 0^{\circ} n

18 0^{\circ} 36 + 36 0^{\circ} n \cdot 36 - 306 0^{\circ}

同类项相加:

648 0^{\circ} - 306 0^{\circ} = 342 0^{\circ}

= 342 0^{\circ} + 2 \cdot 648 0^{\circ} n

数字相乘:

2 \cdot 36 = 72

= 342 0^{\circ} + 1296 0^{\circ} n

= \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

= - \frac{\frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}}{1}

使用分式法则:

\frac{a}{1} = a

= - \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

x = - \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

x = - \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

x = - \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

x = \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{12}, x = - \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

x = \frac{144 0 ^{\circ} n + 18 0 ^{\circ}}{12}, x = - \frac{342 0 ^{\circ} + 1296 0 ^{\circ} n}{36}

作图

流行的例子

-sin(x)-cos(2x)=0

- sin (x) - cos (2 x) = 0

cos(a)= 1/(sqrt(2))

cos (a) = \frac{1}{2}

sin(2θ)-cos(2θ)=0

sin (2 θ) - cos (2 θ) = 0

cos^2(x)+sin(x)=2

cos^{2} (x) + sin (x) = 2

4 csc (θ) - 5 = 0