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受欢迎的三角函数 >

证明 sin^3(x)= 3/4 sin(x)-1/4 sin(3x)

解答

证明

sin^{3} (x) = \frac{3}{4} sin (x) - \frac{1}{4} sin (3 x)

解答

真

求解步骤

sin^{3} (x) = \frac{3}{4} sin (x) - \frac{1}{4} sin (3 x)

调整右侧

\frac{3}{4} sin (x) - \frac{1}{4} sin (3 x)

使用三角恒等式改写

\frac{3}{4} sin (x) - \frac{1}{4} sin (3 x)

利用以下特性:

sin (3 x) = 3 sin (x) - 4 sin^{3} (x)

sin (3 x)

使用三角恒等式改写

sin (3 x)

改写为

= sin (2 x + x)

使用角和恒等式:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

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

使用倍角公式:

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

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

化简

cos (2 x) sin (x) + cos (x) \cdot 2 sin (x) cos (x) : sin (x) cos (2 x) + 2 cos^{2} (x) sin (x)

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

cos (x) \cdot 2 sin (x) cos (x) = 2 cos^{2} (x) sin (x)

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

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

cos (x) cos (x) = cos^{1 + 1} (x)

= 2 sin (x) cos^{1 + 1} (x)

数字相加:

1 + 1 = 2

= 2 sin (x) cos^{2} (x)

= sin (x) cos (2 x) + 2 cos^{2} (x) sin (x)

= sin (x) cos (2 x) + 2 cos^{2} (x) sin (x)

= sin (x) cos (2 x) + 2 cos^{2} (x) sin (x)

使用倍角公式:

cos (2 x) = 1 - 2 sin^{2} (x)

= (1 - 2 sin^{2} (x)) sin (x) + 2 cos^{2} (x) sin (x)

使用毕达哥拉斯恒等式:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= (1 - 2 sin^{2} (x)) sin (x) + 2 (1 - sin^{2} (x)) sin (x)

乘开

(1 - 2 sin^{2} (x)) sin (x) + 2 (1 - sin^{2} (x)) sin (x) : - 4 sin^{3} (x) + 3 sin (x)

(1 - 2 sin^{2} (x)) sin (x) + 2 (1 - sin^{2} (x)) sin (x)

= sin (x) (1 - 2 sin^{2} (x)) + 2 sin (x) (1 - sin^{2} (x))

乘开

sin (x) (1 - 2 sin^{2} (x)) : sin (x) - 2 sin^{3} (x)

sin (x) (1 - 2 sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = sin (x), b = 1, c = 2 sin^{2} (x)

= sin (x) 1 - sin (x) 2 sin^{2} (x)

= 1 sin (x) - 2 sin^{2} (x) sin (x)

化简

1 \cdot sin (x) - 2 sin^{2} (x) sin (x) : sin (x) - 2 sin^{3} (x)

1 sin (x) - 2 sin^{2} (x) sin (x)

1 \cdot sin (x) = sin (x)

1 sin (x)

乘以:

1 \cdot sin (x) = sin (x)

= sin (x)

2 sin^{2} (x) sin (x) = 2 sin^{3} (x)

2 sin^{2} (x) sin (x)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

sin^{2} (x) sin (x) = sin^{2 + 1} (x)

= 2 sin^{2 + 1} (x)

数字相加:

2 + 1 = 3

= 2 sin^{3} (x)

= sin (x) - 2 sin^{3} (x)

= sin (x) - 2 sin^{3} (x)

= sin (x) - 2 sin^{3} (x) + 2 (1 - sin^{2} (x)) sin (x)

乘开

2 sin (x) (1 - sin^{2} (x)) : 2 sin (x) - 2 sin^{3} (x)

2 sin (x) (1 - sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 2 sin (x), b = 1, c = sin^{2} (x)

= 2 sin (x) 1 - 2 sin (x) sin^{2} (x)

= 2 \cdot 1 sin (x) - 2 sin^{2} (x) sin (x)

化简

2 \cdot 1 \cdot sin (x) - 2 sin^{2} (x) sin (x) : 2 sin (x) - 2 sin^{3} (x)

2 \cdot 1 sin (x) - 2 sin^{2} (x) sin (x)

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

2 \cdot 1 sin (x)

数字相乘:

2 \cdot 1 = 2

= 2 sin (x)

2 sin^{2} (x) sin (x) = 2 sin^{3} (x)

2 sin^{2} (x) sin (x)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

sin^{2} (x) sin (x) = sin^{2 + 1} (x)

= 2 sin^{2 + 1} (x)

数字相加:

2 + 1 = 3

= 2 sin^{3} (x)

= 2 sin (x) - 2 sin^{3} (x)

= 2 sin (x) - 2 sin^{3} (x)

= sin (x) - 2 sin^{3} (x) + 2 sin (x) - 2 sin^{3} (x)

化简

sin (x) - 2 sin^{3} (x) + 2 sin (x) - 2 sin^{3} (x) : - 4 sin^{3} (x) + 3 sin (x)

sin (x) - 2 sin^{3} (x) + 2 sin (x) - 2 sin^{3} (x)

对同类项分组

= - 2 sin^{3} (x) - 2 sin^{3} (x) + sin (x) + 2 sin (x)

同类项相加:

- 2 sin^{3} (x) - 2 sin^{3} (x) = - 4 sin^{3} (x)

= - 4 sin^{3} (x) + sin (x) + 2 sin (x)

同类项相加:

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

= - 4 sin^{3} (x) + 3 sin (x)

= - 4 sin^{3} (x) + 3 sin (x)

= - 4 sin^{3} (x) + 3 sin (x)

= 3 sin (x) - 4 sin^{3} (x)

= \frac{3}{4} sin (x) - \frac{1}{4} (3 sin (x) - 4 sin^{3} (x))

化简

\frac{3}{4} sin (x) - \frac{1}{4} (3 sin (x) - 4 sin^{3} (x)) : sin^{3} (x)

\frac{3}{4} sin (x) - \frac{1}{4} (3 sin (x) - 4 sin^{3} (x))

乘开

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

- \frac{1}{4} (3 sin (x) - 4 sin^{3} (x))

使用分配律:

a (b - c) = ab - a c

a = - \frac{1}{4}, b = 3 sin (x), c = 4 sin^{3} (x)

= - \frac{1}{4} \cdot 3 sin (x) - (- \frac{1}{4}) \cdot 4 sin^{3} (x)

使用加减运算法则

- (- a) = a

= - 3 \cdot \frac{1}{4} sin (x) + 4 \cdot \frac{1}{4} sin^{3} (x)

化简

- 3 \cdot \frac{1}{4} sin (x) + 4 \cdot \frac{1}{4} sin^{3} (x) : - \frac{3}{4} sin (x) + sin^{3} (x)

- 3 \cdot \frac{1}{4} sin (x) + 4 \cdot \frac{1}{4} sin^{3} (x)

3 \cdot \frac{1}{4} sin (x) = \frac{3}{4} sin (x)

3 \cdot \frac{1}{4} sin (x)

分式相乘:

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

= \frac{1 \cdot 3}{4} sin (x)

数字相乘:

1 \cdot 3 = 3

= \frac{3}{4} sin (x)

4 \cdot \frac{1}{4} sin^{3} (x) = sin^{3} (x)

4 \cdot \frac{1}{4} sin^{3} (x)

分式相乘:

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

= \frac{1 \cdot 4}{4} sin^{3} (x)

约分:

4

= sin^{3} (x) \cdot 1

乘以:

sin^{3} (x) \cdot 1 = sin^{3} (x)

= sin^{3} (x)

= - \frac{3}{4} sin (x) + sin^{3} (x)

= - \frac{3}{4} sin (x) + sin^{3} (x)

= \frac{3}{4} sin (x) - \frac{3}{4} sin (x) + sin^{3} (x)

同类项相加:

\frac{3}{4} sin (x) - \frac{3}{4} sin (x) = 0

\frac{3}{4} sin (x) - \frac{3}{4} sin (x)

因式分解出通项

sin (x)

= sin (x) (\frac{3}{4} - \frac{3}{4})

\frac{3}{4} - \frac{3}{4} = 0

\frac{3}{4} - \frac{3}{4}

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{3 - 3}{4}

整理后得

= 0

= 0

= sin^{3} (x)

= sin^{3} (x)

= sin^{3} (x)

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

流行的例子

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p ro v e cos^{2} (x) = (1 - sin (x)) (1 + sin (x))

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p ro v e \frac{( cos ( x ) sec ( x ) )}{tan ( x )} = cot (x)

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p ro v e \frac{cot ^{2} ( x ) - 1}{2 cot ( x )} = cot (2 x)

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p ro v e csc (θ) = \frac{sec ( θ )}{tan ( θ )}