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受欢迎的三角函数 >

2sin(6x)sin(3x)=1-cos(9x)

解答

2 sin (6 x) sin (3 x) = 1 - cos (9 x)

解答

x = \frac{2 πn}{3}

+1

度数

x = 0^{\circ} + 12 0^{\circ} n

求解步骤

2 sin (6 x) sin (3 x) = 1 - cos (9 x)

两边减去

1 - cos (9 x)

2 sin (6 x) sin (3 x) - 1 + cos (9 x) = 0

使用三角恒等式改写

- 1 + cos (9 x) + 2 sin (3 x) sin (6 x)

使用积化和差恒等式:

sin (s) sin (t) = \frac{1}{2} (cos (s - t) - cos (s + t))

= - 1 + cos (9 x) + 2 \cdot \frac{1}{2} (cos (3 x - 6 x) - cos (3 x + 6 x))

化简

- 1 + cos (9 x) + 2 \cdot \frac{1}{2} (cos (3 x - 6 x) - cos (3 x + 6 x)) : cos (3 x) - 1

- 1 + cos (9 x) + 2 \cdot \frac{1}{2} (cos (3 x - 6 x) - cos (3 x + 6 x))

2 \cdot \frac{1}{2} (cos (3 x - 6 x) - cos (3 x + 6 x)) = cos (3 x) - cos (9 x)

2 \cdot \frac{1}{2} (cos (3 x - 6 x) - cos (3 x + 6 x))

分式相乘:

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

= \frac{1 \cdot 2 ( cos ( 3 x - 6 x ) - cos ( 3 x + 6 x ) )}{2}

约分:

2

= 1 \cdot (cos (3 x - 6 x) - cos (3 x + 6 x))

同类项相加:

3 x - 6 x = - 3 x

= 1 \cdot (cos (- 3 x) - cos (3 x + 6 x))

同类项相加:

3 x + 6 x = 9 x

= 1 \cdot (cos (- 3 x) - cos (9 x))

整理后得

= cos (- 3 x) - cos (9 x)

使用负角恒等式:

cos (- x) = cos (x)

= cos (3 x) - cos (9 x)

= - 1 + cos (9 x) + cos (3 x) - cos (9 x)

对同类项分组

= cos (9 x) + cos (3 x) - cos (9 x) - 1

同类项相加:

cos (9 x) - cos (9 x) = 0

= cos (3 x) - 1

= cos (3 x) - 1

- 1 + cos (3 x) = 0

将

1

到右边

- 1 + cos (3 x) = 0

两边加上

1

- 1 + cos (3 x) + 1 = 0 + 1

化简

cos (3 x) = 1

cos (3 x) = 1

cos (3 x) = 1

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 = 0 + 2 πn

3 x = 0 + 2 πn

解

3 x = 0 + 2 πn : x = \frac{2 πn}{3}

3 x = 0 + 2 πn

0 + 2 πn = 2 πn

3 x = 2 πn

两边除以

3

3 x = 2 πn

两边除以

3

\frac{3 x}{3} = \frac{2 πn}{3}

化简

x = \frac{2 πn}{3}

x = \frac{2 πn}{3}

x = \frac{2 πn}{3}

作图

流行的例子

cos(θ)sin(θ)=172245

cos (θ) sin (θ) = 172245

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sec (b) cot (b) = sin (b)

(37)/(sin(x))=(30)/(sin(30))

\frac{37}{sin ( x )} = \frac{30}{sin ( 3 0 ^{\circ} )}

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