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受欢迎的三角函数 >

cos(7x)cos(2x)+sin(7x)sin(2x)= 1/2

解答

cos (7 x) cos (2 x) + sin (7 x) sin (2 x) = \frac{1}{2}

解答

x = \frac{π}{15} + \frac{2 πn}{5}, x = \frac{π}{3} + \frac{2 πn}{5}

+1

度数

x = 1 2^{\circ} + 7 2^{\circ} n, x = 6 0^{\circ} + 7 2^{\circ} n

求解步骤

cos (7 x) cos (2 x) + sin (7 x) sin (2 x) = \frac{1}{2}

使用三角恒等式改写

cos (7 x) cos (2 x) + sin (7 x) sin (2 x)

使用角差恒等式:

cos (s) cos (t) + sin (s) sin (t) = cos (s - t)

= cos (7 x - 2 x)

cos (7 x - 2 x) = \frac{1}{2}

cos (7 x - 2 x) = \frac{1}{2}

的通解

cos (x)

周期表（周期为

2 πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

7 x - 2 x = \frac{π}{3} + 2 πn, 7 x - 2 x = \frac{5 π}{3} + 2 πn

7 x - 2 x = \frac{π}{3} + 2 πn, 7 x - 2 x = \frac{5 π}{3} + 2 πn

解

7 x - 2 x = \frac{π}{3} + 2 πn : x = \frac{π}{15} + \frac{2 πn}{5}

7 x - 2 x = \frac{π}{3} + 2 πn

同类项相加:

7 x - 2 x = 5 x

5 x = \frac{π}{3} + 2 πn

两边除以

5

5 x = \frac{π}{3} + 2 πn

两边除以

5

\frac{5 x}{5} = \frac{\frac{π}{3}}{5} + \frac{2 πn}{5}

化简

\frac{5 x}{5} = \frac{\frac{π}{3}}{5} + \frac{2 πn}{5}

化简

\frac{5 x}{5} : x

\frac{5 x}{5}

数字相除:

\frac{5}{5} = 1

= x

化简

\frac{\frac{π}{3}}{5} + \frac{2 πn}{5} : \frac{π}{15} + \frac{2 πn}{5}

\frac{\frac{π}{3}}{5} + \frac{2 πn}{5}

\frac{\frac{π}{3}}{5} = \frac{π}{15}

\frac{\frac{π}{3}}{5}

使用分式法则:

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

= \frac{π}{3 \cdot 5}

数字相乘:

3 \cdot 5 = 15

= \frac{π}{15}

= \frac{π}{15} + \frac{2 πn}{5}

x = \frac{π}{15} + \frac{2 πn}{5}

x = \frac{π}{15} + \frac{2 πn}{5}

x = \frac{π}{15} + \frac{2 πn}{5}

解

7 x - 2 x = \frac{5 π}{3} + 2 πn : x = \frac{π}{3} + \frac{2 πn}{5}

7 x - 2 x = \frac{5 π}{3} + 2 πn

同类项相加:

7 x - 2 x = 5 x

5 x = \frac{5 π}{3} + 2 πn

两边除以

5

5 x = \frac{5 π}{3} + 2 πn

两边除以

5

\frac{5 x}{5} = \frac{\frac{5 π}{3}}{5} + \frac{2 πn}{5}

化简

\frac{5 x}{5} = \frac{\frac{5 π}{3}}{5} + \frac{2 πn}{5}

化简

\frac{5 x}{5} : x

\frac{5 x}{5}

数字相除:

\frac{5}{5} = 1

= x

化简

\frac{\frac{5 π}{3}}{5} + \frac{2 πn}{5} : \frac{π}{3} + \frac{2 πn}{5}

\frac{\frac{5 π}{3}}{5} + \frac{2 πn}{5}

\frac{\frac{5 π}{3}}{5} = \frac{π}{3}

\frac{\frac{5 π}{3}}{5}

使用分式法则:

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

= \frac{5 π}{3 \cdot 5}

数字相乘:

3 \cdot 5 = 15

= \frac{5 π}{15}

约分:

5

= \frac{π}{3}

= \frac{π}{3} + \frac{2 πn}{5}

x = \frac{π}{3} + \frac{2 πn}{5}

x = \frac{π}{3} + \frac{2 πn}{5}

x = \frac{π}{3} + \frac{2 πn}{5}

x = \frac{π}{15} + \frac{2 πn}{5}, x = \frac{π}{3} + \frac{2 πn}{5}

作图

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