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(3x)=3sin(x)cos^2(x)-sin^3(x)

解答

证明

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

解答

真

求解步骤

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

调整左侧

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)

使用倍角公式:

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

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

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

乘开

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

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

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

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

使用指数法则:

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

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

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

数字相加:

1 + 1 = 2

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

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

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

乘开

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

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

使用分配律:

a (b - c) = ab - a c

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

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

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

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

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)

= sin^{2 + 1} (x)

数字相加:

2 + 1 = 3

= sin^{3} (x)

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

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

化简

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

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

对同类项分组

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

同类项相加:

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

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

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

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

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 (cos(3x)-cos(5x))/(sin(3x)+sin(5x))=tan(x)

p ro v e \frac{cos ( 3 x ) - cos ( 5 x )}{sin ( 3 x ) + sin ( 5 x )} = tan (x)

证明 cos^2(x)= 1/2+1/2 cos(2x)

p ro v e cos^{2} (x) = \frac{1}{2} + \frac{1}{2} cos (2 x)

证明 tan(θ/2)=(1-cos(θ))/(sin(θ))

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

证明 1/(2cos(x)sin(x))=csc(2x)

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

证明 cos(x+pi/2)=sin(x)

p ro v e cos (x + \frac{π}{2}) = sin (x)