zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

证明 sin(3A)=3sin(A)-4sin^3(A)

解答

证明

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

解答

真

求解步骤

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

调整左侧

sin (3 A)

使用三角恒等式改写

sin (3 A)

改写为

= sin (2 A + A)

使用角和恒等式:

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

= sin (2 A) cos (A) + cos (2 A) sin (A)

使用倍角公式:

sin (2 A) = 2 sin (A) cos (A)

= cos (2 A) sin (A) + cos (A) 2 sin (A) cos (A)

化简

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

cos (2 A) sin (A) + cos (A) 2 sin (A) cos (A)

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

cos (A) 2 sin (A) cos (A)

使用指数法则:

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

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

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

数字相加:

1 + 1 = 2

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

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

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

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

使用倍角公式:

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

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

使用毕达哥拉斯恒等式:

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

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

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

乘开

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

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

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

乘开

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

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

使用分配律:

a (b - c) = ab - a c

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

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

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

化简

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

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

1 \cdot sin (A) = sin (A)

1 sin (A)

乘以:

1 \cdot sin (A) = sin (A)

= sin (A)

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

2 sin^{2} (A) sin (A)

使用指数法则:

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

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

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

数字相加:

2 + 1 = 3

= 2 sin^{3} (A)

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

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

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

乘开

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

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

使用分配律:

a (b - c) = ab - a c

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

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

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

化简

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

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

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

2 \cdot 1 sin (A)

数字相乘:

2 \cdot 1 = 2

= 2 sin (A)

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

2 sin^{2} (A) sin (A)

使用指数法则:

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

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

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

数字相加:

2 + 1 = 3

= 2 sin^{3} (A)

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

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

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

化简

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

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

对同类项分组

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

同类项相加:

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

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

同类项相加:

sin (A) + 2 sin (A) = 3 sin (A)

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

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

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

我们已展示，在两侧可以有相同的形式

\Rightarrow 真

流行的例子

证明 cot^4(x)+cot^2(x)=cot^2(x)csc^2(x)

p ro v e cot^{4} (x) + cot^{2} (x) = cot^{2} (x) csc^{2} (x)

证明 cot^3(x)=cot(x)(csc^2(x)-1)

p ro v e cot^{3} (x) = cot (x) (csc^{2} (x) - 1)

证明 sin(2a)=(2tan(a))/(1+tan^2(a))

p ro v e sin (2 a) = \frac{2 tan ( a )}{1 + tan ^{2} ( a )}

证明 tan(x)=((1-cos(2x)))/(sin(2x))

p ro v e tan (x) = \frac{( 1 - cos ( 2 x ) )}{sin ( 2 x )}

证明 sin^2(a)-cos^2(a)=1-2cos^2(a)

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