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受欢迎的三角函数 >

证明 sin(4x)=2sin(x)(cos(x)+cos(3x))

解答

证明

sin (4 x) = 2 sin (x) (cos (x) + cos (3 x))

解答

真

求解步骤

sin (4 x) = 2 sin (x) (cos (x) + cos (3 x))

调整左侧

sin (4 x)

使用三角恒等式改写

sin (4 x)

= sin (2 \cdot 2 x)

使用倍角公式:

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

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

使用倍角公式:

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

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

化简

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

使用积化和差恒等式:

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

= 4 sin (x) \frac{1}{2} (cos (2 x - x) + cos (2 x + x))

化简

4 sin (x) \frac{1}{2} (cos (2 x - x) + cos (2 x + x)) : 2 sin (x) (cos (x) + cos (3 x))

4 sin (x) \frac{1}{2} (cos (2 x - x) + cos (2 x + x))

分式相乘:

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

= \frac{1 \cdot 4 sin ( x ) ( cos ( 2 x - x ) + cos ( 2 x + x ) )}{2}

1 \cdot 4 sin (x) (cos (2 x - x) + cos (2 x + x)) = 4 sin (x) (cos (x) + cos (3 x))

1 \cdot 4 sin (x) (cos (2 x - x) + cos (2 x + x))

同类项相加:

2 x - x = x

= 1 \cdot 4 sin (x) (cos (x) + cos (2 x + x))

同类项相加:

2 x + x = 3 x

= 1 \cdot 4 sin (x) (cos (x) + cos (3 x))

数字相乘:

1 \cdot 4 = 4

= 4 sin (x) (cos (x) + cos (3 x))

= \frac{4 sin ( x ) ( cos ( x ) + cos ( 3 x ) )}{2}

数字相除:

\frac{4}{2} = 2

= 2 sin (x) (cos (x) + cos (3 x))

= 2 sin (x) (cos (x) + cos (3 x))

= 2 sin (x) (cos (x) + cos (3 x))

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

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