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受欢迎的三角函数 >

sin(2x)+sin(6x)=2sin(4x)

解答

sin (2 x) + sin (6 x) = 2 sin (4 x)

解答

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

+1

度数

x = 0^{\circ} + 9 0^{\circ} n, x = 4 5^{\circ} + 9 0^{\circ} n, x = 0^{\circ} + 18 0^{\circ} n

求解步骤

sin (2 x) + sin (6 x) = 2 sin (4 x)

两边减去

2 sin (4 x)

sin (2 x) + sin (6 x) - 2 sin (4 x) = 0

使用三角恒等式改写

sin (2 x) + sin (6 x) - 2 sin (4 x)

使用和差化积恒等式:

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

= - 2 sin (4 x) + 2 sin (\frac{2 x + 6 x}{2}) cos (\frac{2 x - 6 x}{2})

2 sin (\frac{2 x + 6 x}{2}) cos (\frac{2 x - 6 x}{2}) = 2 cos (2 x) sin (4 x)

2 sin (\frac{2 x + 6 x}{2}) cos (\frac{2 x - 6 x}{2})

\frac{2 x + 6 x}{2} = 4 x

\frac{2 x + 6 x}{2}

同类项相加:

2 x + 6 x = 8 x

= \frac{8 x}{2}

数字相除:

\frac{8}{2} = 4

= 4 x

= 2 sin (4 x) cos (\frac{2 x - 6 x}{2})

\frac{2 x - 6 x}{2} = - 2 x

\frac{2 x - 6 x}{2}

同类项相加:

2 x - 6 x = - 4 x

= \frac{- 4 x}{2}

使用分式法则:

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

= - \frac{4 x}{2}

数字相除:

\frac{4}{2} = 2

= - 2 x

= 2 sin (4 x) cos (- 2 x)

使用负角恒等式:

cos (- x) = cos (x)

= 2 cos (2 x) sin (4 x)

= - 2 sin (4 x) + 2 cos (2 x) sin (4 x)

- 2 sin (4 x) + 2 cos (2 x) sin (4 x) = 0

分解

- 2 sin (4 x) + 2 cos (2 x) sin (4 x) : 2 sin (4 x) (cos (2 x) - 1)

- 2 sin (4 x) + 2 cos (2 x) sin (4 x)

因式分解出通项

2 sin (4 x)

= 2 sin (4 x) (- 1 + cos (2 x))

2 sin (4 x) (cos (2 x) - 1) = 0

分别求解每个部分

sin (4 x) = 0 or cos (2 x) - 1 = 0

sin (4 x) = 0 : x = \frac{πn}{2}, x = \frac{π}{4} + \frac{πn}{2}

sin (4 x) = 0

sin (4 x) = 0

的通解

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

4 x = 0 + 2 πn, 4 x = π + 2 πn

4 x = 0 + 2 πn, 4 x = π + 2 πn

解

4 x = 0 + 2 πn : x = \frac{πn}{2}

4 x = 0 + 2 πn

0 + 2 πn = 2 πn

4 x = 2 πn

两边除以

4

4 x = 2 πn

两边除以

4

\frac{4 x}{4} = \frac{2 πn}{4}

化简

x = \frac{πn}{2}

x = \frac{πn}{2}

解

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

4 x = π + 2 πn

两边除以

4

4 x = π + 2 πn

两边除以

4

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

化简

x = \frac{π}{4} + \frac{πn}{2}

x = \frac{π}{4} + \frac{πn}{2}

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

cos (2 x) - 1 = 0 : x = πn

cos (2 x) - 1 = 0

将

1

到右边

cos (2 x) - 1 = 0

两边加上

1

cos (2 x) - 1 + 1 = 0 + 1

化简

cos (2 x) = 1

cos (2 x) = 1

cos (2 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}

2 x = 0 + 2 πn

2 x = 0 + 2 πn

解

2 x = 0 + 2 πn : x = πn

2 x = 0 + 2 πn

0 + 2 πn = 2 πn

2 x = 2 πn

两边除以

2

2 x = 2 πn

两边除以

2

\frac{2 x}{2} = \frac{2 πn}{2}

化简

x = πn

x = πn

x = πn

合并所有解

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

作图

流行的例子

cos (x) = \frac{11}{2}

3(cos^2(a))/(sin^2(a))=(cot(a))^2

3 \frac{cos ^{2} ( a )}{sin ^{2} ( a )} = (cot (a))^{2}

cos(x)=(-2)/(sqrt(2))

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

tan (x) = - 0.22

solvefor θ,1+cos(θ)=3cos(θ)

so l v e f or θ, 1 + cos (θ) = 3 cos (θ)