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受欢迎的三角函数 >

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

解答

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

解答

x = πn, x = \frac{π}{2} + πn

+1

度数

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

求解步骤

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

两边减去

2 sin (2 x)

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

使用三角恒等式改写

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

使用倍角公式:

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

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

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

化简

- 2 sin (2 x) + \frac{sin ( 2 x )}{2} : - \frac{3 sin ( 2 x )}{2}

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

将项转换为分式:

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

= \frac{sin ( 2 x )}{2} - \frac{2 sin ( 2 x ) \cdot 2}{2}

因为分母相等，所以合并分式:

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

= \frac{sin ( 2 x ) - 2 sin ( 2 x ) \cdot 2}{2}

sin (2 x) - 2 sin (2 x) \cdot 2 = - 3 sin (2 x)

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

数字相乘:

2 \cdot 2 = 4

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

同类项相加:

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

= - 3 sin (2 x)

= \frac{- 3 sin ( 2 x )}{2}

使用分式法则:

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

= - \frac{3 sin ( 2 x )}{2}

= - \frac{3 sin ( 2 x )}{2}

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

在两边乘以

2

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

在两边乘以

2

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

化简

- 3 sin (2 x) = 0

- 3 sin (2 x) = 0

两边除以

- 3

- 3 sin (2 x) = 0

两边除以

- 3

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

化简

sin (2 x) = 0

sin (2 x) = 0

sin (2 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}

2 x = 0 + 2 πn, 2 x = π + 2 πn

2 x = 0 + 2 πn, 2 x = π + 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

解

2 x = π + 2 πn : x = \frac{π}{2} + πn

2 x = π + 2 πn

两边除以

2

2 x = π + 2 πn

两边除以

2

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

化简

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

x = πn, x = \frac{π}{2} + πn

作图

流行的例子

solvefor n,sin(x)+sin(13 n/2-x)=1

so l v e f or n, sin (x) + sin (13 \frac{n}{2} - x) = 1

5cos^2(a)-2sin(a)-2=0

5 cos^{2} (a) - 2 sin (a) - 2 = 0

tan^2(a)=((2tan(a)))/((1-tan^2(a)))

tan^{2} (a) = \frac{( 2 tan ( a ) )}{( 1 - tan ^{2} ( a ) )}

12cos^2(x)-6=sin(x)

12 cos^{2} (x) - 6 = sin (x)

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