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受欢迎的三角函数 >

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

解答

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

解答

x = 2 πn, x = π + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

sin (x)

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

使用三角恒等式改写

- sin (x) + sin (2 x) tan (x)

使用倍角公式:

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

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

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

分解

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

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

因式分解出通项

sin (x)

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

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

分别求解每个部分

sin (x) = 0 or - 1 + 2 cos (x) tan (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) tan (x) = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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

使用三角恒等式改写

- 1 + 2 cos (x) tan (x)

cos (x) tan (x) = sin (x)

cos (x) tan (x)

使用基本三角恒等式:

tan (x) = \frac{sin ( x )}{cos ( x )}

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

化简

cos (x) \frac{sin ( x )}{cos ( x )} : sin (x)

cos (x) \frac{sin ( x )}{cos ( x )}

分式相乘:

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

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

约分:

cos (x)

= sin (x)

= sin (x)

= - 1 + 2 sin (x)

- 1 + 2 sin (x) = 0

将

1

到右边

- 1 + 2 sin (x) = 0

两边加上

1

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

化简

2 sin (x) = 1

2 sin (x) = 1

两边除以

2

2 sin (x) = 1

两边除以

2

\frac{2 sin ( x )}{2} = \frac{1}{2}

化简

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2}

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 = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

合并所有解

x = 2 πn, x = π + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

作图

流行的例子

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tan(x)(tan(x)-3)+2=0

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2sin(x)=3sqrt(3)

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sin^{2} (θ) = \frac{1}{9}