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cos(7a)=sin(a-6)

解答

cos (7 a) = sin (a - 6)

解答

a = \frac{12 + 4 πn + π}{16}, a = - \frac{π + 4 πn + 12}{12}

+1

度数

a = 54.22183 \dots^{\circ} + 4 5^{\circ} n, a = - 72.29577 \dots^{\circ} - 6 0^{\circ} n

求解步骤

cos (7 a) = sin (a - 6)

使用三角恒等式改写

cos (7 a) = sin (a - 6)

利用以下特性:

cos (x) = sin (\frac{π}{2} - x)

cos (7 a) = sin (\frac{π}{2} - 7 a)

cos (7 a) = sin (\frac{π}{2} - 7 a)

使用反三角函数性质

cos (7 a) = sin (\frac{π}{2} - 7 a)

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

a - 6 = \frac{π}{2} - 7 a + 2 πn, a - 6 = π - (\frac{π}{2} - 7 a) + 2 πn

a - 6 = \frac{π}{2} - 7 a + 2 πn, a - 6 = π - (\frac{π}{2} - 7 a) + 2 πn

a - 6 = \frac{π}{2} - 7 a + 2 πn : a = \frac{12 + 4 πn + π}{16}

a - 6 = \frac{π}{2} - 7 a + 2 πn

将

6

到右边

a - 6 = \frac{π}{2} - 7 a + 2 πn

两边加上

6

a - 6 + 6 = \frac{π}{2} - 7 a + 2 πn + 6

化简

a = \frac{π}{2} - 7 a + 2 πn + 6

a = \frac{π}{2} - 7 a + 2 πn + 6

将

7 a

para o lado esquerdo

a = \frac{π}{2} - 7 a + 2 πn + 6

两边加上

7 a

a + 7 a = \frac{π}{2} - 7 a + 2 πn + 6 + 7 a

化简

8 a = \frac{π}{2} + 2 πn + 6

8 a = \frac{π}{2} + 2 πn + 6

两边除以

8

8 a = \frac{π}{2} + 2 πn + 6

两边除以

8

\frac{8 a}{8} = \frac{\frac{π}{2}}{8} + \frac{2 πn}{8} + \frac{6}{8}

化简

\frac{8 a}{8} = \frac{\frac{π}{2}}{8} + \frac{2 πn}{8} + \frac{6}{8}

化简

\frac{8 a}{8} : a

\frac{8 a}{8}

数字相除:

\frac{8}{8} = 1

= a

化简

\frac{\frac{π}{2}}{8} + \frac{2 πn}{8} + \frac{6}{8} : \frac{12 + 4 πn + π}{16}

\frac{\frac{π}{2}}{8} + \frac{2 πn}{8} + \frac{6}{8}

对同类项分组

= \frac{6}{8} + \frac{2 πn}{8} + \frac{\frac{π}{2}}{8}

使用法则

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

= \frac{6 + 2 πn + \frac{π}{2}}{8}

化简

6 + 2 πn + \frac{π}{2} : \frac{12 + 4 πn + π}{2}

6 + 2 πn + \frac{π}{2}

将项转换为分式:

6 = \frac{6 \cdot 2}{2}, 2 πn = \frac{2 πn 2}{2}

= \frac{6 \cdot 2}{2} + \frac{2 πn \cdot 2}{2} + \frac{π}{2}

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{6 \cdot 2 + 2 πn \cdot 2 + π}{2}

6 \cdot 2 + 2 πn \cdot 2 + π = 12 + 4 πn + π

6 \cdot 2 + 2 πn \cdot 2 + π

数字相乘:

6 \cdot 2 = 12

= 12 + 2 \cdot 2 πn + π

数字相乘:

2 \cdot 2 = 4

= 12 + 4 πn + π

= \frac{12 + 4 πn + π}{2}

= \frac{\frac{12 + 4 πn + π}{2}}{8}

使用分式法则:

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

= \frac{12 + 4 πn + π}{2 \cdot 8}

数字相乘:

2 \cdot 8 = 16

= \frac{12 + 4 πn + π}{16}

a = \frac{12 + 4 πn + π}{16}

a = \frac{12 + 4 πn + π}{16}

a = \frac{12 + 4 πn + π}{16}

a - 6 = π - (\frac{π}{2} - 7 a) + 2 πn : a = - \frac{π + 4 πn + 12}{12}

a - 6 = π - (\frac{π}{2} - 7 a) + 2 πn

展开

π - (\frac{π}{2} - 7 a) + 2 πn : π - \frac{π}{2} + 7 a + 2 πn

π - (\frac{π}{2} - 7 a) + 2 πn

- (\frac{π}{2} - 7 a) : - \frac{π}{2} + 7 a

- (\frac{π}{2} - 7 a)

打开括号

= - (\frac{π}{2}) - (- 7 a)

使用加减运算法则

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

= - \frac{π}{2} + 7 a

= π - \frac{π}{2} + 7 a + 2 πn

a - 6 = π - \frac{π}{2} + 7 a + 2 πn

将

6

到右边

a - 6 = π - \frac{π}{2} + 7 a + 2 πn

两边加上

6

a - 6 + 6 = π - \frac{π}{2} + 7 a + 2 πn + 6

化简

a = π - \frac{π}{2} + 7 a + 2 πn + 6

a = π - \frac{π}{2} + 7 a + 2 πn + 6

将

7 a

para o lado esquerdo

a = π - \frac{π}{2} + 7 a + 2 πn + 6

两边减去

7 a

a - 7 a = π - \frac{π}{2} + 7 a + 2 πn + 6 - 7 a

化简

- 6 a = π - \frac{π}{2} + 2 πn + 6

- 6 a = π - \frac{π}{2} + 2 πn + 6

两边除以

- 6

- 6 a = π - \frac{π}{2} + 2 πn + 6

两边除以

- 6

\frac{- 6 a}{- 6} = \frac{π}{- 6} - \frac{\frac{π}{2}}{- 6} + \frac{2 πn}{- 6} + \frac{6}{- 6}

化简

\frac{- 6 a}{- 6} = \frac{π}{- 6} - \frac{\frac{π}{2}}{- 6} + \frac{2 πn}{- 6} + \frac{6}{- 6}

化简

\frac{- 6 a}{- 6} : a

\frac{- 6 a}{- 6}

使用分式法则:

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

= \frac{6 a}{6}

数字相除:

\frac{6}{6} = 1

= a

化简

\frac{π}{- 6} - \frac{\frac{π}{2}}{- 6} + \frac{2 πn}{- 6} + \frac{6}{- 6} : - \frac{π + 4 πn + 12}{12}

\frac{π}{- 6} - \frac{\frac{π}{2}}{- 6} + \frac{2 πn}{- 6} + \frac{6}{- 6}

对同类项分组

= \frac{π}{- 6} + \frac{6}{- 6} + \frac{2 πn}{- 6} - \frac{\frac{π}{2}}{- 6}

使用法则

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

= \frac{π + 6 + 2 πn - \frac{π}{2}}{- 6}

使用分式法则:

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

= - \frac{π + 6 + 2 πn - \frac{π}{2}}{6}

化简

π + 6 + 2 πn - \frac{π}{2} : \frac{π + 4 πn + 12}{2}

π + 6 + 2 πn - \frac{π}{2}

将项转换为分式:

π = \frac{π 2}{2}, 6 = \frac{6 \cdot 2}{2}, 2 πn = \frac{2 πn 2}{2}

= \frac{π 2}{2} + \frac{6 \cdot 2}{2} + \frac{2 πn \cdot 2}{2} - \frac{π}{2}

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{π 2 + 6 \cdot 2 + 2 πn \cdot 2 - π}{2}

π 2 + 6 \cdot 2 + 2 πn \cdot 2 - π = π + 4 πn + 12

π 2 + 6 \cdot 2 + 2 πn \cdot 2 - π

对同类项分组

= 2 π - π + 2 \cdot 2 πn + 6 \cdot 2

同类项相加:

2 π - π = π

= π + 2 \cdot 2 πn + 6 \cdot 2

数字相乘:

2 \cdot 2 = 4

= π + 4 πn + 6 \cdot 2

数字相乘:

6 \cdot 2 = 12

= π + 4 πn + 12

= \frac{π + 4 πn + 12}{2}

= - \frac{\frac{π + 4 πn + 12}{2}}{6}

化简

\frac{\frac{π + 4 πn + 12}{2}}{6} : \frac{π + 4 πn + 12}{12}

\frac{\frac{π + 4 πn + 12}{2}}{6}

使用分式法则:

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

= \frac{π + 4 πn + 12}{2 \cdot 6}

数字相乘:

2 \cdot 6 = 12

= \frac{π + 4 πn + 12}{12}

= - \frac{π + 4 πn + 12}{12}

a = - \frac{π + 4 πn + 12}{12}

a = - \frac{π + 4 πn + 12}{12}

a = - \frac{π + 4 πn + 12}{12}

a = \frac{12 + 4 πn + π}{16}, a = - \frac{π + 4 πn + 12}{12}

a = \frac{12 + 4 πn + π}{16}, a = - \frac{π + 4 πn + 12}{12}

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