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受欢迎的三角函数 >

sin(3θ+72)=cos(48)

解答

sin (3 θ + 72) = cos (4 8^{\circ})

解答

θ = \frac{1080 0 ^{\circ} n + 126 0 ^{\circ} - 2160}{90}, θ = \frac{414 0 ^{\circ} + 1080 0 ^{\circ} n - 2160}{90}

+1

弧度

θ = \frac{\frac{7 π}{5}}{18} - 24 + \frac{60 π}{90} n, θ = - 24 + \frac{\frac{23 π}{5}}{18} + \frac{60 π}{90} n

求解步骤

sin (3 θ + 72) = cos (4 8^{\circ})

使用三角恒等式改写

cos (4 8^{\circ})

利用以下特性:

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

sin (9 0^{\circ} - 4 8^{\circ})

sin (3 θ + 72) = sin (9 0^{\circ} - 4 8^{\circ})

使用反三角函数性质

sin (3 θ + 72) = sin (9 0^{\circ} - 4 8^{\circ})

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

3 θ + 72 = 9 0^{\circ} - 4 8^{\circ} + 36 0^{\circ} n, 3 θ + 72 = 18 0^{\circ} - (9 0^{\circ} - 4 8^{\circ}) + 36 0^{\circ} n

3 θ + 72 = 9 0^{\circ} - 4 8^{\circ} + 36 0^{\circ} n, 3 θ + 72 = 18 0^{\circ} - (9 0^{\circ} - 4 8^{\circ}) + 36 0^{\circ} n

3 θ + 72 = 9 0^{\circ} - 4 8^{\circ} + 36 0^{\circ} n : θ = \frac{1080 0 ^{\circ} n + 126 0 ^{\circ} - 2160}{90}

3 θ + 72 = 9 0^{\circ} - 4 8^{\circ} + 36 0^{\circ} n

将

72

到右边

3 θ + 72 = 9 0^{\circ} - 4 8^{\circ} + 36 0^{\circ} n

两边减去

72

3 θ + 72 - 72 = 9 0^{\circ} - 4 8^{\circ} + 36 0^{\circ} n - 72

化简

3 θ + 72 - 72 = 9 0^{\circ} - 4 8^{\circ} + 36 0^{\circ} n - 72

化简

3 θ + 72 - 72 : 3 θ

3 θ + 72 - 72

同类项相加:

72 - 72 = 0

= 3 θ

化简

9 0^{\circ} - 4 8^{\circ} + 36 0^{\circ} n - 72 : 36 0^{\circ} n + 4 2^{\circ} - 72

9 0^{\circ} - 4 8^{\circ} + 36 0^{\circ} n - 72

合并分式

9 0^{\circ} - 4 8^{\circ} : 4 2^{\circ}

9 0^{\circ} - 4 8^{\circ}

2, 15

的最小公倍数:

30

2, 15

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

15

质因数分解:

3 \cdot 5

15

15

除以

315 = 5 \cdot 3

= 3 \cdot 5

3, 5

都是质数，因此无法进一步因数分解

= 3 \cdot 5

将每个因子乘以它在

2

或

15

中出现的最多次数

= 2 \cdot 3 \cdot 5

数字相乘:

2 \cdot 3 \cdot 5 = 30

= 30

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

30

对于

9 0^{\circ} :

将分母和分子乘以

15

9 0^{\circ} = \frac{18 0 ^{\circ} 15}{2 \cdot 15} = 9 0^{\circ}

对于

4 8^{\circ} :

将分母和分子乘以

2

4 8^{\circ} = \frac{72 0 ^{\circ} 2}{15 \cdot 2} = 4 8^{\circ}

= 9 0^{\circ} - 4 8^{\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} 15 - 144 0 ^{\circ}}{30}

同类项相加:

270 0^{\circ} - 144 0^{\circ} = 126 0^{\circ}

= 4 2^{\circ}

= 36 0^{\circ} n + 4 2^{\circ} - 72

3 θ = 36 0^{\circ} n + 4 2^{\circ} - 72

3 θ = 36 0^{\circ} n + 4 2^{\circ} - 72

3 θ = 36 0^{\circ} n + 4 2^{\circ} - 72

两边除以

3

3 θ = 36 0^{\circ} n + 4 2^{\circ} - 72

两边除以

3

\frac{3 θ}{3} = \frac{36 0 ^{\circ} n}{3} + \frac{4 2 ^{\circ}}{3} - \frac{72}{3}

化简

\frac{3 θ}{3} = \frac{36 0 ^{\circ} n}{3} + \frac{4 2 ^{\circ}}{3} - \frac{72}{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{4 2 ^{\circ}}{3} - \frac{72}{3} : \frac{1080 0 ^{\circ} n + 126 0 ^{\circ} - 2160}{90}

\frac{36 0 ^{\circ} n}{3} + \frac{4 2 ^{\circ}}{3} - \frac{72}{3}

使用法则

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

= \frac{36 0 ^{\circ} n + 4 2 ^{\circ} - 72}{3}

化简

36 0^{\circ} n + 4 2^{\circ} - 72 : \frac{1080 0 ^{\circ} n + 126 0 ^{\circ} - 2160}{30}

36 0^{\circ} n + 4 2^{\circ} - 72

将项转换为分式:

36 0^{\circ} n = \frac{36 0 ^{\circ} n 30}{30}, 72 = \frac{72 \cdot 30}{30}

= \frac{36 0 ^{\circ} n \cdot 30}{30} + 4 2^{\circ} - \frac{72 \cdot 30}{30}

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 30 + 126 0 ^{\circ} - 72 \cdot 30}{30}

36 0^{\circ} n \cdot 30 + 126 0^{\circ} - 72 \cdot 30 = 1080 0^{\circ} n + 126 0^{\circ} - 2160

36 0^{\circ} n \cdot 30 + 126 0^{\circ} - 72 \cdot 30

数字相乘:

2 \cdot 30 = 60

= 1080 0^{\circ} n + 126 0^{\circ} - 72 \cdot 30

数字相乘:

72 \cdot 30 = 2160

= 1080 0^{\circ} n + 126 0^{\circ} - 2160

= \frac{1080 0 ^{\circ} n + 126 0 ^{\circ} - 2160}{30}

= \frac{\frac{1080 0 ^{\circ} n + 126 0 ^{\circ} - 2160}{30}}{3}

使用分式法则:

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

= \frac{1080 0 ^{\circ} n + 126 0 ^{\circ} - 2160}{30 \cdot 3}

数字相乘:

30 \cdot 3 = 90

= \frac{1080 0 ^{\circ} n + 126 0 ^{\circ} - 2160}{90}

θ = \frac{1080 0 ^{\circ} n + 126 0 ^{\circ} - 2160}{90}

θ = \frac{1080 0 ^{\circ} n + 126 0 ^{\circ} - 2160}{90}

θ = \frac{1080 0 ^{\circ} n + 126 0 ^{\circ} - 2160}{90}

3 θ + 72 = 18 0^{\circ} - (9 0^{\circ} - 4 8^{\circ}) + 36 0^{\circ} n : θ = \frac{414 0 ^{\circ} + 1080 0 ^{\circ} n - 2160}{90}

3 θ + 72 = 18 0^{\circ} - (9 0^{\circ} - 4 8^{\circ}) + 36 0^{\circ} n

将

72

到右边

3 θ + 72 = 18 0^{\circ} - (9 0^{\circ} - 4 8^{\circ}) + 36 0^{\circ} n

两边减去

72

3 θ + 72 - 72 = 18 0^{\circ} - (9 0^{\circ} - 4 8^{\circ}) + 36 0^{\circ} n - 72

化简

3 θ + 72 - 72 = 18 0^{\circ} - (9 0^{\circ} - 4 8^{\circ}) + 36 0^{\circ} n - 72

化简

3 θ + 72 - 72 : 3 θ

3 θ + 72 - 72

同类项相加:

72 - 72 = 0

= 3 θ

化简

18 0^{\circ} - (9 0^{\circ} - 4 8^{\circ}) + 36 0^{\circ} n - 72 : 18 0^{\circ} - 4 2^{\circ} + 36 0^{\circ} n - 72

18 0^{\circ} - (9 0^{\circ} - 4 8^{\circ}) + 36 0^{\circ} n - 72

化简

9 0^{\circ} - 4 8^{\circ} : 4 2^{\circ}

9 0^{\circ} - 4 8^{\circ}

2, 15

的最小公倍数:

30

2, 15

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

15

质因数分解:

3 \cdot 5

15

15

除以

315 = 5 \cdot 3

= 3 \cdot 5

3, 5

都是质数，因此无法进一步因数分解

= 3 \cdot 5

将每个因子乘以它在

2

或

15

中出现的最多次数

= 2 \cdot 3 \cdot 5

数字相乘:

2 \cdot 3 \cdot 5 = 30

= 30

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

30

对于

9 0^{\circ} :

将分母和分子乘以

15

9 0^{\circ} = \frac{18 0 ^{\circ} 15}{2 \cdot 15} = 9 0^{\circ}

对于

4 8^{\circ} :

将分母和分子乘以

2

4 8^{\circ} = \frac{72 0 ^{\circ} 2}{15 \cdot 2} = 4 8^{\circ}

= 9 0^{\circ} - 4 8^{\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} 15 - 144 0 ^{\circ}}{30}

同类项相加:

270 0^{\circ} - 144 0^{\circ} = 126 0^{\circ}

= 4 2^{\circ}

= 18 0^{\circ} - 4 2^{\circ} + 36 0^{\circ} n - 72

3 θ = 18 0^{\circ} - 4 2^{\circ} + 36 0^{\circ} n - 72

3 θ = 18 0^{\circ} - 4 2^{\circ} + 36 0^{\circ} n - 72

3 θ = 18 0^{\circ} - 4 2^{\circ} + 36 0^{\circ} n - 72

两边除以

3

3 θ = 18 0^{\circ} - 4 2^{\circ} + 36 0^{\circ} n - 72

两边除以

3

\frac{3 θ}{3} = 6 0^{\circ} - \frac{4 2 ^{\circ}}{3} + \frac{36 0 ^{\circ} n}{3} - \frac{72}{3}

化简

\frac{3 θ}{3} = 6 0^{\circ} - \frac{4 2 ^{\circ}}{3} + \frac{36 0 ^{\circ} n}{3} - \frac{72}{3}

化简

\frac{3 θ}{3} : θ

\frac{3 θ}{3}

数字相除:

\frac{3}{3} = 1

= θ

化简

6 0^{\circ} - \frac{4 2 ^{\circ}}{3} + \frac{36 0 ^{\circ} n}{3} - \frac{72}{3} : \frac{414 0 ^{\circ} + 1080 0 ^{\circ} n - 2160}{90}

6 0^{\circ} - \frac{4 2 ^{\circ}}{3} + \frac{36 0 ^{\circ} n}{3} - \frac{72}{3}

对同类项分组

= 6 0^{\circ} - \frac{72}{3} + \frac{36 0 ^{\circ} n}{3} - \frac{4 2 ^{\circ}}{3}

使用法则

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

= \frac{18 0 ^{\circ} - 72 + 36 0 ^{\circ} n - 4 2 ^{\circ}}{3}

化简

18 0^{\circ} - 72 + 36 0^{\circ} n - 4 2^{\circ} : \frac{414 0 ^{\circ} + 1080 0 ^{\circ} n - 2160}{30}

18 0^{\circ} - 72 + 36 0^{\circ} n - 4 2^{\circ}

将项转换为分式:

18 0^{\circ} = 18 0^{\circ}, 72 = \frac{72 \cdot 30}{30}, 36 0^{\circ} n = \frac{36 0 ^{\circ} n 30}{30}

= 18 0^{\circ} - \frac{72 \cdot 30}{30} + \frac{36 0 ^{\circ} n \cdot 30}{30} - 4 2^{\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} 30 - 72 \cdot 30 + 36 0 ^{\circ} n \cdot 30 - 126 0 ^{\circ}}{30}

18 0^{\circ} 30 - 72 \cdot 30 + 36 0^{\circ} n \cdot 30 - 126 0^{\circ} = 414 0^{\circ} + 1080 0^{\circ} n - 2160

18 0^{\circ} 30 - 72 \cdot 30 + 36 0^{\circ} n \cdot 30 - 126 0^{\circ}

对同类项分组

= 540 0^{\circ} - 126 0^{\circ} + 2 \cdot 540 0^{\circ} n - 72 \cdot 30

同类项相加:

540 0^{\circ} - 126 0^{\circ} = 414 0^{\circ}

= 414 0^{\circ} + 2 \cdot 540 0^{\circ} n - 72 \cdot 30

数字相乘:

2 \cdot 30 = 60

= 414 0^{\circ} + 1080 0^{\circ} n - 72 \cdot 30

数字相乘:

72 \cdot 30 = 2160

= 414 0^{\circ} + 1080 0^{\circ} n - 2160

= \frac{414 0 ^{\circ} + 1080 0 ^{\circ} n - 2160}{30}

= \frac{\frac{414 0 ^{\circ} + 1080 0 ^{\circ} n - 2160}{30}}{3}

使用分式法则:

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

= \frac{414 0 ^{\circ} + 1080 0 ^{\circ} n - 2160}{30 \cdot 3}

数字相乘:

30 \cdot 3 = 90

= \frac{414 0 ^{\circ} + 1080 0 ^{\circ} n - 2160}{90}

θ = \frac{414 0 ^{\circ} + 1080 0 ^{\circ} n - 2160}{90}

θ = \frac{414 0 ^{\circ} + 1080 0 ^{\circ} n - 2160}{90}

θ = \frac{414 0 ^{\circ} + 1080 0 ^{\circ} n - 2160}{90}

θ = \frac{1080 0 ^{\circ} n + 126 0 ^{\circ} - 2160}{90}, θ = \frac{414 0 ^{\circ} + 1080 0 ^{\circ} n - 2160}{90}

作图

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