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受欢迎的三角函数 >

cos(3x)cos(9x)+sin(3x)sin(9x)=0,(0,2pi)

解答

cos (3 x) cos (9 x) + sin (3 x) sin (9 x) = 0, (0, 2 π)

解答

x = \frac{π}{4}, x = \frac{π}{12}, x = \frac{7 π}{12}, x = \frac{5 π}{12}, x = \frac{11 π}{12}, x = \frac{3 π}{4}, x = \frac{5 π}{4}, x = \frac{13 π}{12}

+1

度数

x = 4 5^{\circ}, x = 1 5^{\circ}, x = 10 5^{\circ}, x = 7 5^{\circ}, x = 16 5^{\circ}, x = 13 5^{\circ}, x = 22 5^{\circ}, x = 19 5^{\circ}

求解步骤

cos (3 x) cos (9 x) + sin (3 x) sin (9 x) = 0, (0, 2 π)

使用三角恒等式改写

cos (3 x) cos (9 x) + sin (3 x) sin (9 x)

使用角差恒等式:

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

= cos (3 x - 9 x)

cos (3 x - 9 x) = 0

cos (3 x - 9 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}

3 x - 9 x = \frac{π}{2} + 2 πn, 3 x - 9 x = \frac{3 π}{2} + 2 πn

3 x - 9 x = \frac{π}{2} + 2 πn, 3 x - 9 x = \frac{3 π}{2} + 2 πn

解

3 x - 9 x = \frac{π}{2} + 2 πn : x = - \frac{π}{12} - \frac{πn}{3}

3 x - 9 x = \frac{π}{2} + 2 πn

同类项相加:

3 x - 9 x = - 6 x

- 6 x = \frac{π}{2} + 2 πn

两边除以

- 6

- 6 x = \frac{π}{2} + 2 πn

两边除以

- 6

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

化简

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

化简

\frac{- 6 x}{- 6} : x

\frac{- 6 x}{- 6}

使用分式法则:

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

= \frac{6 x}{6}

数字相除:

\frac{6}{6} = 1

= x

化简

\frac{\frac{π}{2}}{- 6} + \frac{2 πn}{- 6} : - \frac{π}{12} - \frac{πn}{3}

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

\frac{\frac{π}{2}}{- 6} = - \frac{π}{12}

\frac{\frac{π}{2}}{- 6}

使用分式法则:

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

= - \frac{\frac{π}{2}}{6}

使用分式法则:

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

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

= - \frac{π}{2 \cdot 6}

数字相乘:

2 \cdot 6 = 12

= - \frac{π}{12}

= - \frac{π}{12} + \frac{2 πn}{- 6}

\frac{2 πn}{- 6} = - \frac{πn}{3}

\frac{2 πn}{- 6}

使用分式法则:

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

= - \frac{2 πn}{6}

约分:

2

= - \frac{πn}{3}

= - \frac{π}{12} - \frac{πn}{3}

x = - \frac{π}{12} - \frac{πn}{3}

x = - \frac{π}{12} - \frac{πn}{3}

x = - \frac{π}{12} - \frac{πn}{3}

解

3 x - 9 x = \frac{3 π}{2} + 2 πn : x = - \frac{π}{4} - \frac{πn}{3}

3 x - 9 x = \frac{3 π}{2} + 2 πn

同类项相加:

3 x - 9 x = - 6 x

- 6 x = \frac{3 π}{2} + 2 πn

两边除以

- 6

- 6 x = \frac{3 π}{2} + 2 πn

两边除以

- 6

\frac{- 6 x}{- 6} = \frac{\frac{3 π}{2}}{- 6} + \frac{2 πn}{- 6}

化简

\frac{- 6 x}{- 6} = \frac{\frac{3 π}{2}}{- 6} + \frac{2 πn}{- 6}

化简

\frac{- 6 x}{- 6} : x

\frac{- 6 x}{- 6}

使用分式法则:

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

= \frac{6 x}{6}

数字相除:

\frac{6}{6} = 1

= x

化简

\frac{\frac{3 π}{2}}{- 6} + \frac{2 πn}{- 6} : - \frac{π}{4} - \frac{πn}{3}

\frac{\frac{3 π}{2}}{- 6} + \frac{2 πn}{- 6}

\frac{\frac{3 π}{2}}{- 6} = - \frac{π}{4}

\frac{\frac{3 π}{2}}{- 6}

使用分式法则:

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

= - \frac{\frac{3 π}{2}}{6}

使用分式法则:

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

\frac{\frac{3 π}{2}}{6} = \frac{3 π}{2 \cdot 6}

= - \frac{3 π}{2 \cdot 6}

数字相乘:

2 \cdot 6 = 12

= - \frac{3 π}{12}

约分:

3

= - \frac{π}{4}

= - \frac{π}{4} + \frac{2 πn}{- 6}

\frac{2 πn}{- 6} = - \frac{πn}{3}

\frac{2 πn}{- 6}

使用分式法则:

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

= - \frac{2 πn}{6}

约分:

2

= - \frac{πn}{3}

= - \frac{π}{4} - \frac{πn}{3}

x = - \frac{π}{4} - \frac{πn}{3}

x = - \frac{π}{4} - \frac{πn}{3}

x = - \frac{π}{4} - \frac{πn}{3}

x = - \frac{π}{12} - \frac{πn}{3}, x = - \frac{π}{4} - \frac{πn}{3}

在

(0, 2 π)

范围内的解

x = \frac{π}{4}, x = \frac{π}{12}, x = \frac{7 π}{12}, x = \frac{5 π}{12}, x = \frac{11 π}{12}, x = \frac{3 π}{4}, x = \frac{5 π}{4}, x = \frac{13 π}{12}

作图

流行的例子

1+cot^2(x)+2csc(x)=0

1 + cot^{2} (x) + 2 csc (x) = 0

4sin(2x)=3sin(x)

4 sin (2 x) = 3 sin (x)

sin (a) = 0.1245

solvefor x,sin(x+z)=cos(y+z)

so l v e f or x, sin (x + z) = cos (y + z)

solvefor y,tan(y)= 94/152 cm

so l v e f or y, tan (y) = \frac{94}{152} c m