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受欢迎的三角函数 >

2sin(x)=sqrt(2)sin(2x)

解答

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

解答

x = 2 πn, x = π + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

2 sin (2 x)

2 sin (x) - 2 sin (2 x) = 0

使用三角恒等式改写

2 sin (x) - sin (2 x) 2

使用倍角公式:

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

= 2 sin (x) - 2 \cdot 2 sin (x) cos (x)

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

分解

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

2 sin (x) - 2 cos (x) sin (x) 2

改写为

= 1 \cdot 2 sin (x) - 2 sin (x) cos (x) 2

因式分解出通项

2 sin (x)

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

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

分别求解每个部分

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

1 - 2 cos (x) = 0 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

1 - 2 cos (x) = 0

将

1

到右边

1 - 2 cos (x) = 0

两边减去

1

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

化简

- 2 cos (x) = - 1

- 2 cos (x) = - 1

两边除以

- 2

- 2 cos (x) = - 1

两边除以

- 2

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

化简

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

化简

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

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

使用分式法则:

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

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

约分:

2

= cos (x)

化简

\frac{- 1}{- 2} : \frac{2}{2}

\frac{- 1}{- 2}

使用分式法则:

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

= \frac{1}{2}

\frac{1}{2}

有理化:

\frac{2}{2}

\frac{1}{2}

乘以共轭根式

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

使用根式运算法则:

a a = a

22 = 2

= 2

= \frac{2}{2}

= \frac{2}{2}

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

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

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

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

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

合并所有解

x = 2 πn, x = π + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

作图

流行的例子

sin(θ)+cos(θ)= 7/5 ,sin(2θ)

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tan (x) = \frac{5}{9}

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tan (t) = - \frac{11}{5}, cos (t) > 0, sin (t)

tan(θ+20)tan(90-3θ)=1

tan (θ + 2 0^{\circ}) tan (9 0^{\circ} - 3 θ) = 1

2csc(x)+7/(cos(x))=0

2 csc (x) + \frac{7}{cos ( x )} = 0