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受欢迎的三角函数 >

sin(θ)=cos(2θ+60)

解答

sin (θ) = cos (2 θ + 6 0^{\circ})

解答

θ = \frac{216 0 ^{\circ} n + 18 0 ^{\circ}}{18}, θ = - \frac{90 0 ^{\circ} + 216 0 ^{\circ} n}{6}

+1

弧度

θ = \frac{π}{18} + \frac{12 π}{18} n, θ = - \frac{5 π}{6} - \frac{12 π}{6} n

求解步骤

sin (θ) = cos (2 θ + 6 0^{\circ})

使用三角恒等式改写

sin (θ) = cos (2 θ + 6 0^{\circ})

利用以下特性:

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

sin (θ) = sin (9 0^{\circ} - (2 θ + 6 0^{\circ}))

sin (θ) = sin (9 0^{\circ} - (2 θ + 6 0^{\circ}))

使用反三角函数性质

sin (θ) = sin (9 0^{\circ} - (2 θ + 6 0^{\circ}))

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

θ = 9 0^{\circ} - (2 θ + 6 0^{\circ}) + 36 0^{\circ} n, θ = 18 0^{\circ} - (9 0^{\circ} - (2 θ + 6 0^{\circ})) + 36 0^{\circ} n

θ = 9 0^{\circ} - (2 θ + 6 0^{\circ}) + 36 0^{\circ} n, θ = 18 0^{\circ} - (9 0^{\circ} - (2 θ + 6 0^{\circ})) + 36 0^{\circ} n

θ = 9 0^{\circ} - (2 θ + 6 0^{\circ}) + 36 0^{\circ} n : θ = \frac{216 0 ^{\circ} n + 18 0 ^{\circ}}{18}

θ = 9 0^{\circ} - (2 θ + 6 0^{\circ}) + 36 0^{\circ} n

展开

9 0^{\circ} - (2 θ + 6 0^{\circ}) + 36 0^{\circ} n : - 2 θ + 36 0^{\circ} n + 3 0^{\circ}

9 0^{\circ} - (2 θ + 6 0^{\circ}) + 36 0^{\circ} n

- (2 θ + 6 0^{\circ}) : - 2 θ - 6 0^{\circ}

- (2 θ + 6 0^{\circ})

打开括号

= - (2 θ) - (6 0^{\circ})

使用加减运算法则

+ (- a) = - a

= - 2 θ - 6 0^{\circ}

= 9 0^{\circ} - 2 θ - 6 0^{\circ} + 36 0^{\circ} n

化简

9 0^{\circ} - 2 θ - 6 0^{\circ} + 36 0^{\circ} n : - 2 θ + 36 0^{\circ} n + 3 0^{\circ}

9 0^{\circ} - 2 θ - 6 0^{\circ} + 36 0^{\circ} n

对同类项分组

= - 2 θ + 36 0^{\circ} n + 9 0^{\circ} - 6 0^{\circ}

2, 3

的最小公倍数:

6

2, 3

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

3

质因数分解:

3

3

3

是质数，因此无法因数分解

= 3

将每个因子乘以它在

2

或

3

中出现的最多次数

= 2 \cdot 3

数字相乘:

2 \cdot 3 = 6

= 6

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

6

对于

9 0^{\circ} :

将分母和分子乘以

3

9 0^{\circ} = \frac{18 0 ^{\circ} 3}{2 \cdot 3} = 9 0^{\circ}

对于

6 0^{\circ} :

将分母和分子乘以

2

6 0^{\circ} = \frac{18 0 ^{\circ} 2}{3 \cdot 2} = 6 0^{\circ}

= 9 0^{\circ} - 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{18 0 ^{\circ} 3 - 18 0 ^{\circ} 2}{6}

同类项相加:

54 0^{\circ} - 36 0^{\circ} = 18 0^{\circ}

= - 2 θ + 36 0^{\circ} n + 3 0^{\circ}

= - 2 θ + 36 0^{\circ} n + 3 0^{\circ}

θ = - 2 θ + 36 0^{\circ} n + 3 0^{\circ}

将

2 θ

para o lado esquerdo

θ = - 2 θ + 36 0^{\circ} n + 3 0^{\circ}

两边加上

2 θ

θ + 2 θ = - 2 θ + 36 0^{\circ} n + 3 0^{\circ} + 2 θ

化简

3 θ = 36 0^{\circ} n + 3 0^{\circ}

3 θ = 36 0^{\circ} n + 3 0^{\circ}

两边除以

3

3 θ = 36 0^{\circ} n + 3 0^{\circ}

两边除以

3

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

化简

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

化简

\frac{3 θ}{3} : θ

\frac{3 θ}{3}

数字相除:

\frac{3}{3} = 1

= θ

化简

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

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

使用法则

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

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

化简

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

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

将项转换为分式:

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

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

数字相乘:

2 \cdot 6 = 12

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

= \frac{\frac{216 0 ^{\circ} n + 18 0 ^{\circ}}{6}}{3}

使用分式法则:

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

= \frac{216 0 ^{\circ} n + 18 0 ^{\circ}}{6 \cdot 3}

数字相乘:

6 \cdot 3 = 18

= \frac{216 0 ^{\circ} n + 18 0 ^{\circ}}{18}

θ = \frac{216 0 ^{\circ} n + 18 0 ^{\circ}}{18}

θ = \frac{216 0 ^{\circ} n + 18 0 ^{\circ}}{18}

θ = \frac{216 0 ^{\circ} n + 18 0 ^{\circ}}{18}

θ = 18 0^{\circ} - (9 0^{\circ} - (2 θ + 6 0^{\circ})) + 36 0^{\circ} n : θ = - \frac{90 0 ^{\circ} + 216 0 ^{\circ} n}{6}

θ = 18 0^{\circ} - (9 0^{\circ} - (2 θ + 6 0^{\circ})) + 36 0^{\circ} n

展开

18 0^{\circ} - (9 0^{\circ} - (2 θ + 6 0^{\circ})) + 36 0^{\circ} n : 18 0^{\circ} + 2 θ - 3 0^{\circ} + 36 0^{\circ} n

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

乘开

9 0^{\circ} - (2 θ + 6 0^{\circ}) : - 2 θ + 3 0^{\circ}

9 0^{\circ} - (2 θ + 6 0^{\circ})

- (2 θ + 6 0^{\circ}) : - 2 θ - 6 0^{\circ}

- (2 θ + 6 0^{\circ})

打开括号

= - (2 θ) - (6 0^{\circ})

使用加减运算法则

+ (- a) = - a

= - 2 θ - 6 0^{\circ}

= 9 0^{\circ} - 2 θ - 6 0^{\circ}

化简

9 0^{\circ} - 2 θ - 6 0^{\circ} : - 2 θ + 3 0^{\circ}

9 0^{\circ} - 2 θ - 6 0^{\circ}

对同类项分组

= - 2 θ + 9 0^{\circ} - 6 0^{\circ}

2, 3

的最小公倍数:

6

2, 3

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

3

质因数分解:

3

3

3

是质数，因此无法因数分解

= 3

将每个因子乘以它在

2

或

3

中出现的最多次数

= 2 \cdot 3

数字相乘:

2 \cdot 3 = 6

= 6

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

6

对于

9 0^{\circ} :

将分母和分子乘以

3

9 0^{\circ} = \frac{18 0 ^{\circ} 3}{2 \cdot 3} = 9 0^{\circ}

对于

6 0^{\circ} :

将分母和分子乘以

2

6 0^{\circ} = \frac{18 0 ^{\circ} 2}{3 \cdot 2} = 6 0^{\circ}

= 9 0^{\circ} - 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{18 0 ^{\circ} 3 - 18 0 ^{\circ} 2}{6}

同类项相加:

54 0^{\circ} - 36 0^{\circ} = 18 0^{\circ}

= - 2 θ + 3 0^{\circ}

= - 2 θ + 3 0^{\circ}

= 18 0^{\circ} - (- 2 θ + 3 0^{\circ}) + 36 0^{\circ} n

- (- 2 θ + 3 0^{\circ}) : 2 θ - 3 0^{\circ}

- (- 2 θ + 3 0^{\circ})

打开括号

= - (- 2 θ) - (3 0^{\circ})

使用加减运算法则

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

= 2 θ - 3 0^{\circ}

= 18 0^{\circ} + 2 θ - 3 0^{\circ} + 36 0^{\circ} n

θ = 18 0^{\circ} + 2 θ - 3 0^{\circ} + 36 0^{\circ} n

将

2 θ

para o lado esquerdo

θ = 18 0^{\circ} + 2 θ - 3 0^{\circ} + 36 0^{\circ} n

两边减去

2 θ

θ - 2 θ = 18 0^{\circ} + 2 θ - 3 0^{\circ} + 36 0^{\circ} n - 2 θ

化简

- θ = 18 0^{\circ} - 3 0^{\circ} + 36 0^{\circ} n

- θ = 18 0^{\circ} - 3 0^{\circ} + 36 0^{\circ} n

两边除以

- 1

- θ = 18 0^{\circ} - 3 0^{\circ} + 36 0^{\circ} n

两边除以

- 1

\frac{- θ}{- 1} = \frac{18 0 ^{\circ}}{- 1} - \frac{3 0 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1}

化简

\frac{- θ}{- 1} = \frac{18 0 ^{\circ}}{- 1} - \frac{3 0 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1}

化简

\frac{- θ}{- 1} : θ

\frac{- θ}{- 1}

使用分式法则:

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

= \frac{θ}{1}

使用法则

\frac{a}{1} = a

= θ

化简

\frac{18 0 ^{\circ}}{- 1} - \frac{3 0 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1} : - \frac{90 0 ^{\circ} + 216 0 ^{\circ} n}{6}

\frac{18 0 ^{\circ}}{- 1} - \frac{3 0 ^{\circ}}{- 1} + \frac{36 0 ^{\circ} n}{- 1}

使用法则

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

= \frac{18 0 ^{\circ} - 3 0 ^{\circ} + 36 0 ^{\circ} n}{- 1}

使用分式法则:

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

= - \frac{18 0 ^{\circ} - 3 0 ^{\circ} + 36 0 ^{\circ} n}{1}

化简

18 0^{\circ} - 3 0^{\circ} + 36 0^{\circ} n : \frac{90 0 ^{\circ} + 216 0 ^{\circ} n}{6}

18 0^{\circ} - 3 0^{\circ} + 36 0^{\circ} n

将项转换为分式:

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

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

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} 6 - 18 0 ^{\circ} + 36 0 ^{\circ} n \cdot 6}{6}

18 0^{\circ} 6 - 18 0^{\circ} + 36 0^{\circ} n \cdot 6 = 90 0^{\circ} + 216 0^{\circ} n

18 0^{\circ} 6 - 18 0^{\circ} + 36 0^{\circ} n \cdot 6

同类项相加:

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

= 90 0^{\circ} + 2 \cdot 108 0^{\circ} n

数字相乘:

2 \cdot 6 = 12

= 90 0^{\circ} + 216 0^{\circ} n

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

= - \frac{\frac{90 0 ^{\circ} + 216 0 ^{\circ} n}{6}}{1}

使用分式法则:

\frac{a}{1} = a

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

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

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

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

θ = \frac{216 0 ^{\circ} n + 18 0 ^{\circ}}{18}, θ = - \frac{90 0 ^{\circ} + 216 0 ^{\circ} n}{6}

θ = \frac{216 0 ^{\circ} n + 18 0 ^{\circ}}{18}, θ = - \frac{90 0 ^{\circ} + 216 0 ^{\circ} n}{6}

作图

流行的例子

tan(θ)=(-12)/5

tan (θ) = \frac{- 12}{5}

(1+tan(x/2))/(1-tan(x/2))=sqrt(4.137131)

\frac{1 + tan ( \frac{x}{2} )}{1 - tan ( \frac{x}{2} )} = 4.137131

cos (θ) = 0.51

csc(θ)=(-2sqrt(3))/3

csc (θ) = \frac{- 2 3}{3}

cos (x) = \frac{32}{50}