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受欢迎的三角函数 >

cos(5x)cos(x)+sin(5x)sin(x)=0

解答

cos (5 x) cos (x) + sin (5 x) sin (x) = 0

解答

x = \frac{π}{8} + \frac{πn}{2}, x = \frac{3 π}{8} + \frac{πn}{2}

+1

度数

x = 22. 5^{\circ} + 9 0^{\circ} n, x = 67. 5^{\circ} + 9 0^{\circ} n

求解步骤

cos (5 x) cos (x) + sin (5 x) sin (x) = 0

使用三角恒等式改写

cos (5 x) cos (x) + sin (5 x) sin (x)

使用角差恒等式:

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

= cos (5 x - x)

cos (5 x - x) = 0

cos (5 x - x) = 0

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}

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

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

解

5 x - x = \frac{π}{2} + 2 πn : x = \frac{π}{8} + \frac{πn}{2}

5 x - x = \frac{π}{2} + 2 πn

同类项相加:

5 x - x = 4 x

4 x = \frac{π}{2} + 2 πn

两边除以

4

4 x = \frac{π}{2} + 2 πn

两边除以

4

\frac{4 x}{4} = \frac{\frac{π}{2}}{4} + \frac{2 πn}{4}

化简

\frac{4 x}{4} = \frac{\frac{π}{2}}{4} + \frac{2 πn}{4}

化简

\frac{4 x}{4} : x

\frac{4 x}{4}

数字相除:

\frac{4}{4} = 1

= x

化简

\frac{\frac{π}{2}}{4} + \frac{2 πn}{4} : \frac{π}{8} + \frac{πn}{2}

\frac{\frac{π}{2}}{4} + \frac{2 πn}{4}

\frac{\frac{π}{2}}{4} = \frac{π}{8}

\frac{\frac{π}{2}}{4}

使用分式法则:

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

= \frac{π}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{π}{8}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

约分:

2

= \frac{πn}{2}

= \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

解

5 x - x = \frac{3 π}{2} + 2 πn : x = \frac{3 π}{8} + \frac{πn}{2}

5 x - x = \frac{3 π}{2} + 2 πn

同类项相加:

5 x - x = 4 x

4 x = \frac{3 π}{2} + 2 πn

两边除以

4

4 x = \frac{3 π}{2} + 2 πn

两边除以

4

\frac{4 x}{4} = \frac{\frac{3 π}{2}}{4} + \frac{2 πn}{4}

化简

\frac{4 x}{4} = \frac{\frac{3 π}{2}}{4} + \frac{2 πn}{4}

化简

\frac{4 x}{4} : x

\frac{4 x}{4}

数字相除:

\frac{4}{4} = 1

= x

化简

\frac{\frac{3 π}{2}}{4} + \frac{2 πn}{4} : \frac{3 π}{8} + \frac{πn}{2}

\frac{\frac{3 π}{2}}{4} + \frac{2 πn}{4}

\frac{\frac{3 π}{2}}{4} = \frac{3 π}{8}

\frac{\frac{3 π}{2}}{4}

使用分式法则:

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

= \frac{3 π}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{3 π}{8}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

约分:

2

= \frac{πn}{2}

= \frac{3 π}{8} + \frac{πn}{2}

x = \frac{3 π}{8} + \frac{πn}{2}

x = \frac{3 π}{8} + \frac{πn}{2}

x = \frac{3 π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}, x = \frac{3 π}{8} + \frac{πn}{2}

作图

流行的例子

sin(θ)=(5n+1)/3-2n

sin (θ) = \frac{5 n + 1}{3} - 2 n

cosh (x) - 1 = 0

arctan(θ)=arctan(92/44)

arctan (θ) = arctan (\frac{92}{44})

10cos(x)-5sqrt(2)=0

10 cos (x) - 52 = 0

sin(2x)=sin^2(2x)

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